HOW T0 BECOME 


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THE UNIVERSITY 
OF ILLINOIS 
LIBRARY 


The 
Frank Hall collection 
of arithmetics, 
presented by Professor 
H. L. Rietz of the 
Universit 


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Lee Xx SMe A 
EMATICS LIBRARY ay eS ARI | F 
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HOW TO BECOME 


QUICK 


AT 


FIGURES. 


——-COMPRISING — 


THE SHORTEST, QUICKEST, AND BEST METHODS OF BUSI- 
NESS CALCULATIONS. 


e/a ee GONE: 


MPitLOor, Sl00 FPOsTPATD. 
For Sale by all Book and Newsdealers. 


1885. 
BOSTON, MASS: 


THE WOODBURY COMPANY, 
PUBLISHERS. 


at 


brarian’s Office 


the Li 


in 


Washing ton, 1883. 


Entered according to act of Congress 


ML 


aA SSe 2\ 


MEN WANTED. 


“It has been truly said that the great want of the 
age is men. Men of thought; men of action. Men 
who are not for sale. ‘Men who are honest to the 
heart’s core. Men who will condemn wrong in friend 
or foe —in themselves as well as others. Men whose 
consciences are as steady as the needle to the pole. 
Men who will stand for the right if the heavens totter 
and the earth reels. Men who can tell the truth and 
look the world and the devil right in the eye. Men 
who neither swagger nor flinch. Men who are Quick 
at Figures. Men who can have courage without 
whistling for it, and joy without shouting to bring it. 
Men through whom the current of everlasting life 
runs still, and deep and strong. Men too large for 
cer tain limits, and too strong for sectarian bands. 
Men who know their message and tell it. Men who 
know their duty and do it. Men who know their 
place and fill it. Men who mind their own business. 
Men who will not lie. Men who are not too lazy to 
work, nor too proud to be poor. When in office, the 
workshop, in the counting-room, in the bank, in 
every place of trust and responsibility, we can have 
such men as these, we shall have a christian civiliza- 
tion — the highest and best the world ever saw.” 


464239 


CONTENTS. 


PAGE 

Abbreviations in Prescr apsions, ; : 2 : : 107 
Addition, ; , F . : TOT, 
Drill tables, . ; < : - i11—17 

“e From left to right, ; r : A 17 

as General rules, : ; ‘4 ri ° : oT 

$6 Grouping, : A : y 2 . 10 

es Ledger columns, . : “ . 2 22 

6 Lightning agers : . - ‘ : 19—22 

66 Results only, ‘ : ; : . 8 

“ the Easy method, 8 “ z 4 f 23—26 

‘s with periods, < ‘ ae ; 4 ; 26 

“6 2 columns at once, . ; ‘ : is 15 

<e 3 columns at once, . : - : Q 18 

Ale and Beer Measure, . ‘ 4 4 } i 106 
Aliquot Parts, application of ° gta Fa . . 45 
Amusing Arithmetic, . ° : : : ‘ 131—144 
Apothecary’s Weight, : : , F ; 106 
Avoirdupois Weight, : : . : : - 95 
Banks, Transactions with . , ‘ A 5 . 121—122 
Bills of Exchange, . : - 4 t . 76 
Brick work, . ae . A : . ; 113 
Brokers’ Technicalities, : A ; : . ; 122 
Butter, . < : ; F : : 99 
Calculations for particular branches, . ; ‘ ‘ 78 
Calculating Rates on Nails, Ss . p - 119: 
Capacity of Cisterns per foot, . > : - 5 105 
Carpenters’ Estimates, : ‘ : : : ; 115 
Check, How to Endorse . : Sgt, ee: ‘ Ve 
Cisterns, round A : “ : : . - 104—105 
Cisterns, square. : : : ; : 104 
Clapboards, . - : : ‘ A 115 
Coal, How to Estimate in Bulk . : . ; 100 
Coins of of oclids Nations, . ; : F . T2—T74 
Contents of S 2 ; - ; 4 : 112 
Corn in Crib, . * ; : ‘ - p 108 
Cubic Measure, . ; : 4 ; : 112 
Day of Week, How to abies n : ‘ : : A 128 
Decimals, . ; ; : Pe : 67—68 
Division, Contractions : é : : ; - 53—54 
Do Something, ; ° : : : : - 6 
Drafts and Acceptances, 3 : : ‘ : : 76 
Dry Measure, 4 ; ° : i : : 108 
English to U. S. Money, . p : : 80 
Fractions, . 4 : n ; , : : 55—66 
és Addition, . ; ; ; : : ; 57 

BE Contractions, . A : ‘ : : 60—66 

a Division, . A ; ‘ < ‘ 58 

cM Mixed N umbers, : A ‘ ‘ : 63 

es Multiplication, : ‘ ‘ , A F 57 

J Relation to . zs - A ; : 58 

ie Subtraction, ; - : Z “ 57 

#6 To a Common Denominator, ; : 4 56 

ss To Lower ee: ‘ : 2 56 
Freight, R.R. 2 mn ‘ ‘ 5 * 102 
Great Britain’s Money, : : ; : - A 74 
Hay, To Estimate . : - A : 4 . 96—98 
Interest, 5 : - : ? 4 ‘ - 81—94 
“ Bankers’ Method, : : - ; : 85 


ee by Cancellation, . : : ; : : 87 


CONTENTS — CONTINUED. 


Interest, Common Method, r 
for Days only, 
“More or Less than 60 Days, 
ES Partial Payments, . 


BS to find the Principal, . : ° 

< to find the Rate, : “ 

s&s to find the Time, é A . 

“6 Vermont Rule, < A A 


ie 6 per cent. Method, . 
ee $12 Rule, or Lightning Method, 


Land Measure, : . : i 
Laths, . : : 

Length of Nails, : 

Liquid Measure, - : ° 

Long Measure, - : - “ 

Long Ton Weight, 


Mariner’s Measure, ; 
Marking Goods, 
Masonry, . ° 
Measures, : 
Metric System, F 
Multiplication, . ; 
aS Aliquot Parts, . - 
Bo Contractions, 3 - 
as Cross Method, . 
és Sliding Method, 
se Squaring, 
6c when the Tens are © Alike, 4 
Multiples, Table of 


Nails, 
Calculating Rates, ‘ 
« Length 
“s Meaning, ‘at Penny, 
s¢ Number in a Pound, 
Notes, Description of A 


. 
. 


Paper, Names of various styles, 
Particular Branches, 4 : = 
Perches, A 

66 How to Estimate 

“6 Short Method, 
Printers’ Table, . : 4 , 
Publishers’ Table, Z - 3 A 


Rates on Nails, how to calculate F 
Round Cisterns, ; ! : ; 


Shingles, to Estimate. J 
Shoemakers’ Measure, ; : 
Silver Coins, ; . 

Square Measure, . 

State Currency, A 
Surveyors’ Measure, . . 


The Day of the Week, - 
Troy Weight, . . : . 


U.S. Coins, : ; J ; 
U.S. Money, . ‘ : : 


Valuable Information, . : 
Ls eg ny echt 


128—130 
105 


70 
69 


75 


95 
99 


DO SOMETHING. 


Do not spend your precious time in wishing, and watch- 
ing, and waiting for something to turn up. If you do, you 
may wish and watch and wait forever. You can do it if 
you wish, but you must put forth the effort. Idleness and 
indifference never accomplished anything. It takes energy 
and push to make headway in the world, and an active, 
energetic, persevering man is sure to succeed. If he can 
not do one thing he will do something else. If he can not 
succeed in one direction he will in some other. He will 
do something. He will not waste his time in idleness. 
There is no lack of work, no lack of opportunities. Do 
what comes to your hand, and doit well. True progress 
is from the less tothe greater. You must begin low if you 
would build high. Work is ordinarily the measure of suc- 
cess. Quit resolving and re-resolving and go and do 
something. — School Supplement. 


ADDITION. 


The adding of one or more columns of figures 
should be done without mental labor, and may be 
acquired by anyone with a good deal of practice. 


The art of adding quickly is acquired by learning 
to vead a column of figures as you would a sentence 
composed of words, and those words composed of 
letters. By Practice we have become so familiar with 
letters that when we see them grouped together, it is 
unnecessary to separate them, or spell out the words, 
but we can tell at a glance what the word is. 


By Practice we may become so familiar with figures 
that when we see a group of them, we can tell ata 
glance what the sum of them reads, without spelling 
the figures at all. 

In practicing the reading of a column of figures in 
this way, we do not let the brain work at all, but 
simply pass the eye over the figures (see drill tables) 
as if you were reading a sentence, slowly at first, but 
increase the speed as proficiency is acquired. 


A few minutes daily practice will produce astonish- 
ing results in a very short time ; beginning with two 
figures, then three, four, and so on until finally we 
become able to write the Swm total of long columns. 
For example, when we see the figures 9, 8, 6, 4, we 
know at a glance that the sum is 27 without reading 
the figures themselves or spelling them out. 


Reading a column of figures as the reading of a 
sentence, is done by dividing a large group of figures 


ADDITION. 


into smaller ones and from group to group through 
the column, just as from word to word we read 
through a sentence. 

We give various methods, but commend the gvoup- 
ing method as the best and most practiced. 

Addition is more frequently used than all other 
operations combined. 

The most important qualities of an accountant are 
accuracy and speed. The most speedy calculators 
are usually the most correct. 

It is a deplorable fact that not one in one hundred 
of our graduates fresh from the High School can 
add a column of figures correctly without many trials. 

No labor should be regarded too great to master 
this, the key to all numerical as well as business 
transactions. 


RESULTS ONLY. 


Never spell your way through a column, thus: 
6 and 8 are 14, and 9 are 23, etc. It is just as easy 
to name results only, and much more rapid. 

For the purpose of explaining a method, examples 
will be sometimes spelled out, but it is never recom- 
mended as a method to be adopted. 


367 Begin at the bottom of the right hand 
854 column, and name results only: 
976 14, 23, 31, 37, 46, 52, 56, 63. 


389 Then adding 6, the carrying figure, to the 


736 second column, we have: 

j ; i 15, 17, 24, 28, 31, 39, 46, 51, 57. 

726 Again carrying 5 to the next columr we 
98 SY: 


11, 18, 22, 27, 34, 37, 46, 54, 57, 
© 7 «3 which completes the operation. 


NotTre.—When 9 occurs in addition it is easier to add ten and sub- 
ract one mentally. thus: instead of 9+ 8=17, say, 10-+8=18—1=—17. 


ADDITION. 


GROUPING. 


SPELLING PROCESS. 
145. 11+10+4+410+3-4 10 = 48 
Perea tO 8 10 2 Tg 10 Demi 
be 5 10 1-110 11011-45104 Sen 


The sum of any two figures can never exceed a 
ten; therefore, the left hand figure of the sum of 
any two figures must always be one. 


It is, by this method, no more difficult to read the 
sum of two figures, than to read a single figure. 


In the process we pay no attention to the figures 
as they stand written, but only the sum of two of 
them. 


In the above example, instead of reading 11, 20, 
25, 33, etc., we say: | 


11, 21, 25, 35, 38, 48. 
4, 14, 17, 27, 29, 39, 49, 51. 
5, 15, 16, 26, 36, 37, 47, 50. 


It will be noticed that the left han figure 
increases by regular notation, 1, 2, 3, 4, 5, 6, etc. 


Practice will enable one to add much more rapidly 
than by the old method, with less liability to error, 
because it is systematic and simple, and requires less 
‘mental labor. 


10 


GROUPING. 


By this method we mentally group the figures 
above 10 and under 20. 

To the first group add the tens of the next group, 
and to this sum add the units of the second group. 

In order to become proficient in grouping, famil- 
iarity of totals will be absolutely necessary. The sum 
of the figures must be read instead of the figures them- 
selves. By a proper understanding of this method, 
the mental labor is much less than by the old ones ; 
half the labor is simply counting by regular notation, 
because the left hand or tens figure to be added is 
always the same, and the right hand or units figure 
usually of a small denomination. By an analysis of 
the example under this head, we find the first col- 
umn in group to read :— 
7 ; 10 To 11, the first group, add the tens of 
3 the second group, we have 21, adding the 
aM 13 unit 4 of this group we have 25, adding the 
5 ) tens of the third group we have 35, and add: 
3 14 ing 8, the units figure of the group we have . 
at 11 238, adding the tens of the fourth group we 
3 have 48 which completes the operation. 

SECOND COLUMN. 


4 12 4 the carrying figure of the previous col- 
a ns umn, added to the first group we have :— 


4+10+3+10+24-10+1012=51 

4 

3 12 By naming result only, we have :— 
: 13. 4,14, 17, 27, 29, 89, 49, 51 


We recommend to name the carrying figure the 
first, and when a nine occurs call it 10, and subtract 


lin the operation. 


1] 


ADDITION. 


DRILL TABLES. 


These Drill Tables must be thoroughly mastered. 
Read from left to right and right to left, the sum 
only, as rapidly as possible ; when you falter or make: 
an error, go back and start again. 


We would advise teachers to have a daily black- 
board exercise of these drill tables for addition. 


Divide the exercises. Write series that are sim- 
ple and easily comprehended. 


Ist. Read the sum of two figures only, in every: 
possible combination. 


2d. Read the sum of three figures only. 


3d. Read in two columns of two figures, then 
three, and also mixed numbers. 


It will make a pleasant, entertaining, as well as 
useful change from the regular routine. One month’s. 
daily exercise will result in surprising proficiency. 


The whole school should be engaged in these exer- 
cises, and combinations varied and made more diffi- 
cult, slowly. 


Hfint. One of the most accomplished accountants 
in the city of Boston, informed the writer that he 
owed his success as an accountant entirely to his 
being a confident adder, which he acquired largely by 
appropriating all figures in sight, and reading their 
sum only. Numbers on the streets, numbers of 
railroad cars, etc., etc., served as drill masters to him 
on many occasions. 


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DRILL TABLE No, 1. 


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ADDITION. 


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ADDITION. 


DRILL TABLE No, 2. 


Ani Sins ae a Ag 
Sieh ie Oy ae aaa D 
SLA A Bat ea! oy 
Lene Quince gn SSA 
Gey Ose Ads Gt: 7 
an MEV as OER 
Gin Aad be Suis cine 8 
ie Ciba Ma 
Ger Sees ule Tenn () 
uh Said. 1 Ow 8 
Sh Giicd ny Ging 
Ga Se ae Sins 
Gi Fee Gyaw Biel ft 
sind Wht hes ioc elon 
Tat Ox ae oes 
Subas Been Gamay Skt G 
Pity Gah Sali? 
Sie Gate Givi Qa ware 
SO AP ivS eG 
Asi aie eR ake Rs ae 
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13 


14 


ADDITION, 


DRILL TABLE No. 3. 


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15 


TWO COLUMNS AT ONE OPERATION, 


FROM LEFT TO RIGHT 


If two or more columns can be added at one oper- 
ation, there must be some rule or method by which 
it is done. The following illustrates one of the 
most practical methods :— 


2 5 SPELLING PROCESS. 

63 27-+50=77-14— 81 

78 81-30 = 1b oS 

34 115+ 70 =185+8 — 193 

54 193 +. 60 = 258-13 = 256 

2 7 256-20 == 276.1 281 
281 


This process consists simply in adding the tens 
first and then the units. 


BEADING PROCESS. 


27, 77, 81, 115, 185, 193, 253, 256, 276, 281. 


Practice will enable one to add two or more col- 
umns without much mental effort, because it is just 
as easy to say, 27 +50 — 77-+- 4 — 81, as it is to 
say, 5+2=—7, or 7+ 4=11. 


Proficiency in the above method will soon enable 
one to add without separating each number into 
tens and units, thus :— 


27, 81, 115, 193, 256, 281. 


The all important thing is familiarity of totals, 
and we herewith append drill tables, applicable to 
this kind of addition, which must be read from 
each direction as advised before. 


16 


ADDITION. 


DRILL TABLE No. 4. 


43 52 
25 34 
64 25 
52 38. 
63 89 
94 98 
38 68 
46 89 
93 38 
36 48 
97 39 
84 98 
78 63 
87 36 
85 95 
98 79 
63 Ya) 


84 
21 


he 


ADDITION. 


DRILL TABLES. 


Read rapidly from left to right and right to left 
the sum only in the following combinations : — 


SEE NO, HOM. 2 Oi Gis Om LOL dat) Oy Vas 
Bete ihe 01 6S) Oy oA EOL COUN On Ban Onn alec d 
24 35 72 94 39 36 28 
32 46 82 38 48 59 39 
Daim aoe km Octane SOONG Merete ah Gh eG 
Dear AO Oca” ih ) LOR! Oc OY ES 
PEDRO A Oe) Oe Oem On aie O Rd 


Oe ee OE) 


1.25 12.50 9°50 29.40 45.90 
84 3.42 21 3.50 1.50 
5.62 40 4.21 6.30 56.89 


Combinations should be varied and made more 
difficult as proficiency is acquired. The secret in 
adding rapidly consists in familiarity with totals of 
combinations. Counting is not adding, and spelling 
is not reading. 


18 


ADDITION, 
THREE COLUMNS AT ONE OPERATION, 
FROM RIGHT TO LEFT, 


Three or more columns mar be added at one op 
eration thus :— 
223 
425 
384 
256 


1288 
OPERATION.— 256-+-4—=260, 260-+-80=340, 340-+- 
300—640, 640-+-5—645, 645+-20—665, 665+400— 
1065, 1065-+-3—1068, 1068+4-20=1088, 1088-+-200 
=1288, 


By naming results only, we have :— 
260, 3840, 640, 
645, 665, 1065, 
1068, 1088, 1288. 


It will be noticed that beyond two, or at most. 
three, columns wide and a fewcolumns deep, this 
method requires more mental labor than the previ- 
ous ones, and is not considered very practical except 
for development of the mental faculties. 


EXAMPLE 2. 

2.25 

.385 
8.46 PROCESS. 

-08 340, 350, 353, 1853, 1853, 1918, 1921, 

15.60 2721, 2761, 2767, 2797, 2802, 

2.13 3002, 3022, 3027. 
1.40 


19 
ADDITION. 


THE LIGHTNING METHOD. 


AN OLD TRICK EXPOSED. 


This simple and wonderful combination of figures 
has deluded many into believing that adding from 
left to right such a large body of figures, instantane- 
ously and correctly, to be a Herculean task. 

Large sums have been paid by the unsuspecting 
and credulous for the possession of the wonderful 
secret. 

It is probably the simplest, as well as the most 
delusive combination of numbers, because if opera- 
ted in the hands of an expert it is almost impossible 
to be detected, unless by those who know the secret. 


EXAMPLE: 
ay PEGs BTg ST Mtoe mak fete bites ene: Seep 4 
Gr roiaan oie be Lt Oi Grn ad 
SPAY Vite Oe: DONS a rata: ULES MIE Bibs 
Gay shia cp Geel venue Oa Lin O 
Fi hts at cin ts RUNS ehube tiie: 8 ta yee Che aK | 
PLING Oost 2 OG an Ott Ono 


The operation is as follows: A line of figures is 
given you, to the length of which you are wholly in- 
different, as in above example. 


ered 00 OF USN) Soke Gel,e 4s. (Bizet ting 


Now you claim the privilege of writing a line im. 
mediately underneath it. Your line will be in pairs 
of 9’s, and will read: 


6 ly 4 3 1 1 0) 6 5 / (Second line.) 


Now «sk another line immediately under these ; 
any figures whatever, for example: 


3 5 8 4 8 8 4 8 8 , Y (Third Line. 


20 
ADDITION. 


You write another to pair it into 9’s: 
66 40512 868 46241455°.6 2o1%,.0) fourth Line) 


Another line of figures is given you: 
78 6 (8) 90 AUiey Rist tiie 


This is called the key line, because the sum of the 
entire column is simply this line repeated, with 2 
subtracted from the right, and 2 annexed to the left. 
In other words, the result being just like it, except 
the units’ place, which is as many less than the units 
in the key line as there are pairs of lines, and annex 
a similar number to the extreme left of the sum. 
The number of lines is necessarily odd. 


For the purpose of: explanation we will add the 
column without the key line. 


BF: Marts lm ones Whitt Fak NS colts Satie: Bolo 
OD ee Omak eas Oe OU paltn| 
SO Bae Sk Br Lh Oly Chane 
GSA sa aes ohne 00s Gemea aan 
| ane Bis Basie ae irs kiN wes Dn aes ee melee en (eget 

2 
yan UA Vs ODAC RE aed Pie TAU RR ie 


Subtracting two from the right hand figure of the 
key line is equivalent to adding two to the other col- 
umns, which would bring all the other figures ciphers 
as above. 


It matters not, therefore, what the key line is, so 
long as we call the right hand figure two less, and 
annex two to the left. 


A little practice will make this a ve’y interesting 
trick. 


21 
ADDITION. 


SECOND APPLICATION OF THE “LIGHTNING” METHOD. 


Take any three columns of figures as follows :— 


63456 
38429 
25636 


Pair the last two with 9’s and all the columns will 
stand thus :— 


63456 
38429 
61570 
25636 
74363 


263454 


The top line repeated with 2 subtracted from the 
_ right hand figure and 2 annexed on the left, com- 
pletes the operation. 


THIRD APPLICATION OF THE LIGHTNING METHOD: 


8987. Key Line. 


—$——$—$—<$— 


48983 


In this example we have four pairs of nines; 
therefore, subtract 4 from the right hand figure in the 
key line, and annex 4 on the left. 


22 


ADDITION. 


LEDGER COLUMNS. 


A great part of the work of an accountant consists 
in adding long ledger columns, like the following. 
Let the pupil find the sum of the numbers in each, 
being as careful to obtain a correct result as he 
would be if he were to receive or pay the several 
amounts. 


Combinations should be varied and made more 
difficult as proficiency is acquired. The secret in 
adding rapidly consists in familiarity with totals of 
combinations. Counting is not adding, and spelling 
is not reading. 


8.37 18 673.28 
4.353 AT 597.84 
7.62 ys) 3426.87 
48 2.795 219.48 
Od 1.20 8.37 
2.50 4.37 167,84 
6.19 8.29 5986.32 
10.00 13.85 6749.31 
4.28 2.00 4863.27 
8.07 62 1542.35 
4.37 20 2986.28 
9.48 1.37 379.87 
4.21 9.83 2.99 
13.26 6.75 69.80 
1.20 8.43 4060.75 
ot %).48 309.71 
3.08 6.00 124.87 
4.96 1.00 8520.06 
85 1.50 2493.28 


4.00 (.69 48.75 


THE EASY WAY TO ADD. 


This method of adding numbers is especially de- 
signed for those whose minds are constantly em- 
ployed with business affairs, the mind being relieved 
at intervals, and the mental labor of carrying over 
the sum of an entire column is obviated by the use 
of /ntegers or “catch figures.” 


86 


EXAMPLE. 


Process.—Begin at 9 to add as near 20 as you 
can, thus: 9-+ 2+ 4-+ 3= 18, reject the 
tens and place the 8 to the right of the 3, as in 
example ; begin at 6 and add 6+ 8-+-4= 18, 
reject the tens, as before, and place 8 to the 
right of 4, as in example; begin at 6 +- 7 4-4 
= 17, reject tens, place 7 to the right of 4, 
as in example; then 9 + 4+ 3= 16, reject 
tens, place 6 to the right of 3; then 6 + 7+ 
4 = 17, reject tens and place 7 to the right, as 
before ; having arrived at the top of the column, 
add the figures in the new column, thus: 8-+ 8 
+7-+ 6-+ 7=36, or 3 tens and 6 units ; place 
the 6 units as the unit’s figure of the sum, hav- 


ing 3 tens to carry to.5 tens, the number of integers 
or catch figures already rejected. 3 + 5 = 8 tens, 
which prefixed with the 6 makes 86 the sum. 


24 


ADDITION. 


N. B.—The small figures we set to the right are called 
integers, ‘‘tally,” or catch figures. 

If upon arriving at the top of the column there 
should be one or more figures whose sum will not 
equal 10, add them to the sum of the figures in the 
new column ; never place an extra figure in the new 
column unless it is an excess over 10. 


3 EXAMPLE.—Proceed as before ; begin at the 
bottom, 2-+ 7-+- 6 = 15, reject the tens, place 
3s 5 to the right of 6; 6+ 4-+5=— 15, reject as 
7 before, place 5 to the right of 5; say 8 + 7 +- 
8 3-== 18, reject as before, etc. Now we have 
5° three figures which do not add 10; add them 
; to the new column and say 5 + 5-+- 8 + 4-4 
gs 2-+3= 27; place the 7 under the original 
7 column, add 2 to the number of tally figures, 
2 which is 3, thus: 3 -+- 2 =5, the tens figure in 
oi the sum, and makes 57, the answer. 


Two or more columns can be added in the same 
manner. 
6939 EXAMPLE. — Proceed as in adding a single - 
65 column. The sum of the first column being 
49, we write the 9 and carry the 4 to the next 
49s column, thus: 4-- 2-8-3 == 17, reject 
53 the tens as before, write the 7 to the left of the 
765 3; then proceed as with a single column. 
REMARK.—To add very long columns, it is 
734 better to add as near 30 as possible, instead of 
82 20asin above examples. The reason for sug- 
23 gesting this method is to decrease the number 
659 of “tally” figures. It must be remembered, 
however, by adopting 380 as a standard, that two tens 
will be rejected instead of one, as in the former ex. 
amples, which will be observed in the following: 


25 


ADDITION. 


to 


EXAMPLE. 


Process.—Here we begin at the bottom as be- 
fore, adding over 20 and under 30, placing the 
excess or tally figure wherever it occurs in the 
line, thus: 4+5-+4+6+3-+442 = 
28, reject the 2 tens, place the 8 to the right of 
2, etc. 


© Ot OO # bO OS 


a 


By an inspection of the example it will be 
; seen that8-+8-+9-+7-+ 8+ 2= 42, the 
sum of the tally figures ; place 2 as the unit’s 
figure of the sum ; carry 4 to twice the number 
of tally figures, since the twenties, not the tens, 
were rejected as tallies, so we say twice 6 are 
12 and 4 to carry are 16; prefix this with the 2 
units, and we have 162, the sum of the whole 
column. 

Notr. — The reason for adding numbers by tally 
figures must be clear, since it is nothing more than 
a condensing process, which can be briefly explained 
g thus: take for instance the tally figures in the pre- 

ceding example which are 8, 8, 9, 7, 8, 2, which. when 

fully written out, would make the column read, 28 

But since the tens’ figures are all alike, it is 28 

necessary to write the units only, and simply 29 

bear in mind that for every unit’s figure written 27 


out we have two tens, and thus abbreviate. 28 


REMARK.—This mode of adding is espe- ae 


cially designed for those whose minds are con- 
startly employed on business affairs, and who 
are apt to be interrupted. A little practice will 
enable anyone to add rapidly and a/ways cor- 
rectly without any mental labor or fatigue. 
162 But the young accountant, whose dusiness it 
is to add, and xothing else, should rely entirely upon 
the mind and adopt the preceding rules. 


eoOP DWE NWN WRAP ER DOWD LR WONe OSL DAIS ODP AO = 


26 


iT 
S| Co NS Oo RD OO RDI GO OTE BD 


ADDITION. 


Adding with Periods. 


Another condensed method of Addition is 
by periods, which is illustrated in the following 
example: 

Commence thus?)$. 454095273 2 
= 18, reject ten, place a period to the right of 
2, carry 8 to the next figure, thus: 8 3 4 

2 == 17, place the period to the right of 2, 
reject ten, carry 7 to next figure,7 4 3 

4 = 18 ; place period to the right of 4, reject 
ten, .carry $ to next figures (0). ae ee 
19; reject ten, place 9 as the unit’s figure in 
the sum ; the number of periods, which are 4 in 
this case, will be the tens’ figure in the sum of 
the column and completes the addition. 


27 


General Rules for Addition. 


1. Write numbers plainly and distinctly, so 
that 9’s may not be mistaken for 7’s, or 5’s for 3’s. 

2. Write the numbers in vertical lines. _Ir- 
regularity in placing of figures is the cause of many errors. 

3. Think of results, and not of the numbers them- 
selves. Thus do not ue 4 and 5 are 9, and 6 are 15, and 7 
are 22, etc.; but‘9, 15, 22, etc. 

4. Make Mere tinte of 10 or other numbers as 
often as possible, and add them as single numbers. Thus: 


Ce ee 


in adding 9 34 73 214 9 54 82 123, say 9 16 26 33 42 51 
61 67, taking each group at a glance as a single number. 
When a figure is repeated several times, multiply instead 
of adding. 

5. In adding horizontally begin at the left, since 
the eye is more accustomed to moving from left to right 
than from right to left. 

6. In adding long columns, prove the work by 
adding each column separately in the opposite direction 
before adding the next column. 

We believe that addition should be drilled into every boy, 
every day, from the beginning to the end of his school life. 
It is ten times more important than measures, multiples, 
interest, percentages, stocks, etc., all put together, and the 
sooner teachers thoroughly understand this fact the more 
practical and beneficial will be the result ofour school system. 

Pupils should be trained to add figures when placed in a 
horizontal position as well as when placed in a vertical 
position. In the following exercises add both ways, and 
then prove the results by adding them. 


Ex.1. 4621 38946 4256 8432 1562 =— 22817 
421 5000 7060 85 984 — **** 

2012 12138 214 143 O75 = **** 

1604 21038 1524 21388 4215 = **** 

2385 6214 3121 1562 1428 = **** 


11043 RR aio takok 


3 |80]| 214 |20}| 14/20}]) 14 |27 2) 17 
25 2 |60}| 181/10 1 |00}; 13/84 
175 |16 1 |25}| 19/40}/ 125 |10}} 184/15 
10 |80}} 13 )75}) 161)15 2 |00 6)17 


—— |—__ ee ed i ee eee 


Ex. 2. ¢42 |50|/ $13 |40||9243/10]| $3 |o4|l$136 ees | 


1 


MULTIPLICATION. 


In the ordinary process of multiplication we ob- 
tain partial products and then add these together for 
the entire product. With a little wholesome prac- 
tice, however, we multiply by a number consisting 
of several figures, without writing out the partial 
products. There are those who can multiply by a 
number consisting of 10 to 12 or even more digits, 
writing the result under the given number with great 
readiness. This is a very unusual degree of profi- 
ciency ; but almost anyone can learn to do the same 
with a multiplier consisting of from 2 to 6 places. 
We indicate the method by the following problems 
and solutions. To make the operations easily un- 
derstood, we have selected small numbers at first, - 
and advance into higher and more difficult ones, 
step by step, and whoever studies the method 
thoroughly, and practices perseveringly, will be am- 
ply rewarded for the time devoted to the task. 


EXAMPLES. — PROCESS. 
23 


6 


lst. 236, we write down the 6 for the 
______ units figure in the product. 

736 =. 2d. (2 2)-+-(83)=138, we write down the 3 
for the ten’s figure in the product, and reserve 1 to 
carry. 

3d. 32=—6-+1 (we carried) =7, which com- 
pletes the product. 


29 
MULTIPLICATION. 


EXAMPLE 2. 

78 Ist. 7 X 856, we write down 6 as the unit’s 
_____ figure in the product, reserving 5 to carry. 
2106 2d. (7 X7)4+(2X8)}+5 (to carry )—=70; We 
write down the 0 for the ten’s figure in the product, 
reserving 7 to carry. 

3d. 2 X +-+-7=21, which completes the product. 

EXAMPLE 3. 

126 Ist. 5 X 630, write down O as the unit’s 
_____ figure in the product, reserving 3 to carry. 
4410 2d. (5X2)+(3X6)+3=31, write 1 for the 
ten’s figure in the product, reserving 38 to carry. 

3d. (5X 1)+(8 X 2)+3=14, write 4 for the hun- 
dred’s place in the product, reserving 1 to carry. 

4th. 3 X 1+1=—4, which completes the product. 

EXAMPLE 4. 

Ist. 6 X 424, write 4 as the unit’s figure 
in the product, reserve 2 to carry. 7 
141264 2d. (3 X 4)-+(2 X 6)4+-2=26, write 6 as the 
ten’s figure in the product, reserving 2 to carry. 

3d. (6X3)+(4 X 4)+(3 X 2) +-2=—42, write 2 as the 
hundred’s figure in the product, reserving 4 to carry. 

4th. (8 X3)+(4 X 2)+-4—21, write 1 as the thov- 
sand’s figure in the product, reserving 2 to carry. 

5th. 4 X 34+-2—14, which completes the product. 


324 


THE SLIDING METHOD 


MULTIPLICATION: 


Probably the easiest method to learn to multiply | 
large numbers in a single line is the sliding method 
as used by Peter M. Deshong, which is in reality 
nothing more than cross multiplication, as_ illus- 
trated ; but for the beginner it is the best that can 
be adopted. When familiar with the slide the stu- 
dent can proceed without it, and perform operations 
astonishing to himself and those who witness the 
operation, the largest numbers being readily multi- 
plied in a single line. 

This method can easily be understood by following 
the examples and solutions here given with paper 
and pencil. 


31 
MULTIPLICATION. 


EXAMPLE. — PROCESS, 


324 : aay : 
436 Write the multiplier on a slip of paper 


separate from that on which the multipli- 
141264 cand is written, in an inverted order, thus: 
634; place this slip directly over the multiplicand, 
so that the 4 will be directly over the 6, thus: 
643 
324 
then say 6 X 424, write 4 as your unit’s figure in 
the product, reserving 2 to carry ; now slide the pa- 
per to the left so that 2 will come under 6, and 4 
under 3, thus: 
634 
324 
now (6X 2)-+-(4 X 3)+-2==26, write 6 as the ten’s figure, 
reserving 2 to carry ; again slide the paper to the left 
so that 3 falls under 6, 2 under 3, and 4 under 4, 


thus: 634 
324 


and you have (6 X 3)-+-(3 X 2) x (4 X 4)-+-2==42, write 2 
as hundred’s figure in product, reserve 4 to carry; slide 
the paper again and the 3 will be under 3, and 2 un- 
der 4, thus: 634 

324 
and you have (3 X3)-+(4X 2)-++-4=21, write 1 as the 
thousand’s figure in the product, reserving 2 to car- 
ry ; now slide again, that 3 will be under 4, thus: 

634 

324 
and you have 3X4-+2=—14, which completes the 
produs:, 141264. 


We have used the same figures in this, as in the 
preceeding example, and by ciose observation it can 
readily be seen that the work is all the same. The 
sliding method, however, saves the mental labor of 


32 
MULTIPLICATION. 


carrying over, in the mind, so many figures, which is 
quite wearisome to the unpractised mind. ‘These 
additions will soon be performed at a glance, as the 
producis are obvious without the formality of naming 
factors, which the student should never allow him- 
self to do in any operation; it is just as easy to 
name products only. To understand these directions 
thoroughly, factors must be placed upon slips of pa- 
per, and the directions strictly complied with, which 
will give an insight into the mode of operation, and 
the reason will be better understood in ten minutes, 
than in three hours without them. When once fam- 
iliar with the slide, the student may proceed without 
it. We will solve another example upon the same 
principle, naming products only, as it should be oper- 
ated. 
EXAMPLE. — PROCESS. 
5768 
324 On a separate slip of paper, as before, 
invert the multiplier, thus: 423 ; place the 
1868832 multiplier so that 8 will be under 4, and 
you have 32; write 2 for units in product, reserve 3, 
carry 4; slide, say 24-++-16-+-3—43, write 3, carry 4; 
slide, 28--12-+-24--4—68, or thus: 28, 40, 64, 68, 
write 8, carry 6 as before, and slide again, 20, 34, 
52, 58, write 8, carry 5; slide, 10, 31, 36, write 6, . 
carry 3; slide once more, 15, 18: you have the com- 
plete product. 


REMARK.—Proceed towards the left until the multipli- 
cand passes from under the multiplier, each time adding 
what you carry to the sever:l products that stand one over 
the other, and the result will be the product. 


33 


MULTIPLICATION. 


SLIDING METHOD. 


5768 
324 


1,868,832 


425 


4658 


2,901,934 
326 
4658 


326 
4658 


326 
4658 


326 
4658 


326 
4658 


326 
4658 


EXAMPLE FIRST 


Reverse the multiplier, thus: 423. 


Reading products only. 


32 
9441-1643 me 43 
98 112i 94 “Ws 2168 
20-+14-+18+6 = 58 
JO ton G28 86 
Ub igt Geet 8 


EXAMPLE SECOND. 


Reverse the multiplier, thus: 326. 


24 
16-+15 + 2 = 338 
48+10+184+38= 79 
30+124+124+7=61 
36 +8-+ 6 = 50 


24+5=—=29 


CONTRACTIONS 


MULTIPETOCATION 


Contractions can often be advantageously em- 
ployed in business calculations ; but, like by-paths in 
a forest, they are convenient only to those who know . 
the whole ground. Strangers will do better to keep 
the highway. 


TO MULTIPLY BY ELEVEN. 


Ruiz. — Add the figures in the multiplicand, after the 
first, from right to left. 


APPLICATION 1.—45 X11. 4-++-5=9; place this 
sum between 4 and 5, thus 495. 
35 X11. 3+ 5= 8; place this sum between 3 
and 5, thus 385. / 
AD 11 == 462. 63 X 11 = 693. 
oh B78 Be eet 44 X 11 = 484, 
95 "Aili "1045: AO XoUT e589: 


35 


CONTRACTIONS IN MULTIPLICATION. 


APPLICATION 2.—345 X11. Here we write 5; we 
say 4+5= 9. write 9; then4-+3—7; write 7; 
then write 3, thus 3795. 

254 X11. Write 4 for the first figure in the pro- 
duct; 4+ 5= 9, write 9 for the second figure ; 
5 + 2 = 7 which is the third figure, and write 2 for 
the last figure, and we have 2794. 


B20 x LP '3575:; 12 Xl Ba 189 
353 X 11 = 8883. PMD ATG Dyker (LE Yap 


Nore.—If the sum of two figures is over 9, carry the one 
to the next figure. 

APPLICATION 8.—58 X 11 = 638 ; here we write 8 
as the first figure, and say 5 + 8 = 13; write 3 for 
the middle figure, and carry the one to the next fig- 
ure 5, making the product 638. 

TALL o20. SDM O45. 
885 X 11 = 4235. 5863 X 11 = 64493, 


TO MULTIPLY TWO FIGURES BY TWO FIGURES WHEN 
THE TENS ARE ALIKE. 
To multiply 87 by 82. 

Multiply units by units for the first figure of the 
product, the sum of the units by tens for the second 
figure, and tens by tens for the remaining figures, 
carrying when necessary. 

Wir weal AS carry” 1; 
Damo 2 he ON io oe anal ho 


7134 8X 8 = 64, and 7 to carry = 71. 


E-xercises. 
81 X 87 81 X 87 ints, ee ive 
62 X 63 AT ae LOTS 
54 X 55 56 X 52 107 X 105 
ABIX AT 79 X 75 Wns gy, 


27, X22 44 xX 43 113 X 114 


36 
MULTIPLICATION. 


CONTRACTIONS. 
TO SQUARE ANY NUMBER OF NINES. 
Rule, —Write from left to right as many nines, 


less one, as the given number contains, an 8, as 
many ciphers as nines, and 1. 


Thus the 9:9 9 Be rae TO I 
square 9999 99980001 
of 9/1 9>929°9 99998000014 


TO MULTIPLY BY ANY NUMBER OF NINES. 


Rule. — Annex as many ciphers to the right of 
the multiplicand as there are nines in the multiplier, 
and from this number subtract the multiplicand ; 
the remainder will be the product required. 


EXAMPLE. — 37645 & 9999. 


8376412355 


The reason is obvious. By annexing four ciphers, 
we multiply the given number 10000 times; and by 
subtracting the given number, we have the product 
one less than 10000, or 9999 times the number. 


TO MULTIPLY BY ANY NUMBER ENDING IN NINE, 


Rule. — Multiply by the next higher number, and 
subtract the multiplicand. 


EXAMPLE. — 42 X 389. 


39+ 1= 40 
42 « 40 — 1680 minus 42 = 1638.—Azs. 


37 
MULTIPLICATION. 


CONTRACTIONS. 


TO MULTIPLY BY ANY NUMBER FROM TWELVE 
TO TWENTY. 


Rule, — Multiply in regular succession the figures 
of the multiplicand by the unit’s figure of the mul- 
tiplier, and add to the product of each multiplication 
that figure in the multiplicand which stands next on 
the right of the one which you multiply ; add, also, 
the figure to carry, if any. 


EXAMPLE. — 3 6 4 3 5 
13 


473655 


Here we say 8 X 5 = 15; write down 5, carry 1; 
say 3 X 3-++ 1+ 5, the figure which stands on the 
right of 3 = 15; write 5, carry 1; say3 X 4+1 
+ 3 = 16; write 6 and carry 1;3 xX 6+1-+4 
= 23; write 8, carry 2; 3X 38+2+6=17; 
write 7, carry 1 to 38 = 4. 


TO MULTIPLY BY 21, 31, 41, 51, 61, 71, 81, 91. 


Rule. — Write down the units figure of the multi- 
plicand as the first figure of the product. Multiply 
in regular succession every figure in the multiplicand 
by the left hand figure of the multiplier, and to each 
product add the figure which stands next on the left 
of that which you multiply, and you have the requir- 
ed product. 


EXAMPLE: 


3725 Here we write the 5 as the units figure 
21. in the multiplier; then say 5 XK 2 +2 
= 12; write 2, carry 1;say2 KX 2+1+ 
78225 7 = 12; write 2, carryl1; say2 x 7+1 
+ 3 = 18: write 8, carry 1; then say 

¢x 3-1 = 7, which completes the product. 


38 
MULTIPLICATION. 


CONTRACTIONS. 


TO MULTIPLY ANY NUMBERS OF TWO PLACES EACH, 
WHEN THE UNITS OR TENS ARE ALIKE. 


ule. — Multiply units by units ; then, if the units 
are alike, multiply the sum of the tens, and the tens 
by the tens. If the tens are alike, multiply the sum 
of the units by the tens, and the tens by tens; in 
all cases carrying as usual. 


EXAMPLE 1: 

34 4X 4 = 16; write 6, carry 1. 

54 548 x41 — 33; write 3, carry 3. 
—— 5X3+ 3= 18, which completes the 
1836 product. 

EXAMPLE 2: 

45 5 & 8 = 15; write 5, carry 1. 


1 2 
43 5+3 x 4+ 1 = 33; write 3, carry 3. 
3 = 19, completes the product. 
1938 
This rule will apply to the square of any number. 
It is the most useful of all contractions, and should 
be carefully studied. 


TO MULTIPLY BY NUMBERS WHICH ARE FROM ONE 
TO TWELVE LESS THAN ONE HUNDRED, 
ONE THOUSAND, ETC. 

Rule.— Multiply the multiplicand by the differ- 
ence between the multiplier and 100, 1000, &c., and 
subtract the product from the product of the multi- 
plicand by 100, 1000, &c. 

EXAMPLE. — Multiply 35 by 98. 

98 — 100 — 2. 
SD (270 
35 & 100 = 3500, 
3500 — 70 = 3430. — Ans. 

When from 1 to 12 more than 100, add the prod- 
uct of the multiplicand by the unit figure, after 
annexing the required number of ciphers, thus: 


EXAMPLE. — Multiply 325 by 102. 


325 & 100 = 32500. 
325 & 2 = 650 + 32500 = 38150. 


39 


MULTIPLICATION. 


CONTRACTIONS. 


WHEN THE SUM OF THE UNITS IS TEN, AND THE 
TENS ARE ALIKE. 


Method. — Multiply the units and write the result 
as the first two figures in the product. Then call 
the tens figure one more and write their product for 
the last two figures in the final product. 


EXAMPLE, 
86 4xX6=2 
8 4 8-+-1x< 8=7 


7224 


93 85 48 63 56 71 66 
97 85 42 67 54 79 64 


This contraction will only apply where the units 
equal ten and the tens are alike, as in above exam- 
ple. If the product of units does contain ten, as in 
9 X 10, the place of tens must be supplied with a 
cipher. 


TO SQUARE ANY NUMBER ENDING IN FIVE. 


Method, — Multiply the figure preceding the units 
as they will stand by the next higher number, and 
‘ annex 26 to the product. 


EXAMPLE. 

795 T+1ixK7= 56, 

75 Annex 26. 
5625 


WHAT IS THE SQUARE OF 


257 S85? 45? 75? 85? 95? 105? 115? 125? 
185? 145? 155? 165? 175? 185? 195? 205? 


40 


MULTIPLICATION. 
CONTRACTIONS. 


TO FIND THE PRODUCT OF ANY TWO NUMBERS 
WHOSE UNIT FIGURES ARE FIVE. 


* Method. — Take the product of the figures pre- 
ceding the 5 in each number, increase this by one- 
half of the sum of these figures, and prefix the result 
to 25. 


EXAMPLE, 


25 4*%2+3=11. 
45 


1125 
WHAT IS THE VALUE OF 


25x45? 55X75? 7x95? 65> 95? 
385 85? 85X45? 155 X 85: 165 X 45? 
185 X65? 175% 65? 225 % 105? 


Notre. —If the sum of the figures preceding the 5 is odd, when we 
take one-half of it, the one-half or five-tenths which remains must be 
added to the figure 2 of the 25; or we may take one-half of the next 
smaller number, and use 75 as the suffix. 


TO MULTIPLY BY TWO FIGURES AT ONCE. 


Rule. — Multiply both figures in the multiplier by 
each figure in the multiplicand separately. 


Note. — When large numbers are to be multiplied, for the purpose 
of remembering which figure has been used, place a dot over each 
figure of the multiplicand as soon as multiplied. 


. 


EXAMPLE. 
3265 Bx D4 ae 207, 
24 6x 24+12=—156 


(pha ss 2x24115=—63 
78360 3x 2416=78 


To Multiply by Aliquot Parts of 
100, 1000, Ete. 


It is very important for an accountant to have a 
perfect knowledge of the table of Aliquot Parts of 
100 and 1000. All goods sold at wholesale are 
bought and sold by these calculations, and those not 
‘amiliar with the operation will often lose much val- 
uable time in obtaining a correct result. By this 
method they can arrive at the result in one-tenth of 
the time, and are not so apt to make mistakes. 


ALIQUOT PARTS. 


Ororo: 
2=4 49> Ws 
ood St US) 
Of 100. 
6z = 1s 16g = 4 50 == 4 
Slow. 2 = 4 62 — gs 
12,5 = 4 Boa 3 fi) mel? 
Mis} 8a 8m =k 
18¢ = 3 3814 — +, 
Of 1000. 
Soh esp ieee 
bey fer ie 250 \—= 4 375 = 8 


625 == 8 or 1-16 of 10,000. 
8334 = 2 or 1-16 of 1,000. 875 = 


42 


ExaMPLeE 1.— Multiply 464 by 25 = 11600. 
4)46400 
11600 


This is, in effect, the same as to multiply by 100 
we divide by 4 because 25 is 4 of 100, which is th. 
same as multiplying by 25. In the same manner, 
annex two ciphers and divide by 2 multiplies by 50 ; 
annex two ciphers and divide by 8 multiplies by 124 ; 
or annex two ciphers and divide by 8 to multiply by 
125, etc., etc. 

This same principle may be applied in any Aliquot 
Part of 10, 100, 1000, as shown in preceding table. 


RuLE.—Add ciphers to the multiplicand and divide by 
the number, as the multiplier is a part of 100 or 1000. 
When the multiplicand is a mixed number, reduce the 
fraction to a decimal and proceed as before. 


ExampPLeE 1.— Multiply 434 by 24. 


4) 43400 
10850 


EXAMPLE 2.— Multiply 535 by 25. 
4)53500 
13375 


EXAMPLE 3.— Multiply 5642 by 34. 
3)56420 
188062 


EXAMPLE 4.— Multiply 4321 by 334. 
3) 4382100 
1440332 


EXAMPLE 5.— Multiply 1254 by 13. 
6)123840 
20568 


EXAMPLE 


EXAMPLE 


EXAMPLE 


EXAMPLE 


EXAMPLE 


EXAMPLE 


EXAMPLE 


EXAMPLE 


EXAMPLE 


6.— Multiply 2245 by 13. 
7)22450 
32071 


7.— Multiply 4456 by 14. 
8) 44560 
5570 


8.— Multiply 5644 by 14. 
9)56440 


9.— Multiply 4324 by 64. 
16) 432400 
27025 


10.— Multiply 5642 by 84. 
12) 564200 
470162 


11.—Multiply 5648 by 124. 
8) 564800 
70600 


12.— Multiply 6843 by 142. 


7) 684300 
97757} 


13.— Multiply 7824 by 163. 


6) 782400 
130400 


14.— Multiply 7846 by 834. 


12)7846000 
6538334 


43 


44 


Examp.e 15.— Multiply 7896 by 125 
$4of 1000. 8)7896000 
987000 


EXAMPLE 16.— Multiply 1246 by 16632. 
4of 1000. 6)1246000 
2076664 


EXAMPLE 17.— Multiply 8453 by 250. 
$ of 100. 4)8453000 
2113250 


EXAMPLE 18.— Multiply 4642 by 625. 
qs of 10000. 16)46420000 


or, 2901250 

& of 1000. 8) 4642000 
580250 

alas isha niiee a2, 

2901250 


EXAMPLE 19.— Multiply 5642 by 8334, 
vz of 10000. 12)56420000 


or, 47016662 

2 of 1000. §)5642000 
9403334 

i A 
47016663 


EXAMPLE 20.— Multiply 1342 by 875. 
Z0f 1000. 8)1342000 


167750 
7 


1174250 


4D 


APPLICATION OF THE TABLE OF ALIQUT PARTS, 


In order to give an idea of rapid calculation in 
multiplication, a few examples will here be given, 
and from these others may be created without limit 
by anyone: 

35 yards cloth @ $2.50. Add one cipher and divide 
by 4. Answer, 874 or $87.50. 

216 yards cloth @ $2.25. Multiply $24 by sp 
setting down the amounts thus: “$486 


48 yards cloth @ $2.124. Multiply by $24 in same 
manner as by $24. 


55 yards cloth @ $1.95. Move decimal point 97.50 
in price two places to the right, divide 195 9.75 


by 2 and add +, thus: $107.25 
162 yards cloth @ $1.80. Multiply by 2 and eet 
deduct ay thus : $291.60 
36 yards cloth @ $1.75. Multiply by 2 and ta 
deduct 4, thus: $63 
29 
29 yards cloth @ $1.624. Add 4 and to 14.50 
the whole, thus- 3.623 
$47.12 
114,50 
1144 yards cloth @ $1.50. Add 4, thus: 57,25 
$171.75 
37.75 
372? yards cloth @ $1.25. Add 4, thus: 9.44 
$47.19 
83 
83 yards cloth @ $1.20. Add }, thus: 16.60 


$99.60 


46 


APPLICATION OF THE TABLE OF ALIQUOT PARTS. 


50 yards cloth @ $1.18. Find 4 of $118 = $59. 


114 


75 yards cloth @ $1.14. Deduct from $114 28.50 
4 of that amount, thus: $85.50 
24 yards cloth @ 95c. Deduct sh from $24, 74 
yards c c. Deduct 5'5 from $ 1.20 
thus: “$22.80 
68.50 
684 yards cloth @ 75c. Deduct 4, thus: 17.12 
| $51.38 
Eathetiy 23 
46 yards cloth @ 55c. Find $ and add to 2 30 
same ;,, thus: $25.30 
: 16 
32 yards cloth @ 45c. Find 4 and deduct 1.60 
Zo, thus : $14.40 


96 yards cloth @ 25c. Find 4 of $96 = $24. 


The reason for making the computations in this 
manner will at once be apparent, from the fact that 
when the price is either more or less than $1, the 
fractional part of a dollar is either taken from, or 
added to, the amount that sum would be if @ $1 per 
yard. It is sometimes more convenient to call the 
number of yards the price, and the price the number 
of yards, in order to make the computation, as in the 
example above ; 50 yards @ $1.18 would be the same 
as 118 yards @ 50c. ; or to say, if 100 yards @ $1.18 
would be $118, 50 yards would be half of that amount, 
or $59. In the first example we say, if 35 yards @ 
$10 per yard would be $350, at $2.50 per yard it 
would be one-fourth of that amount, or $87.50. 


47 


It is of course expected that the student will per- 
form these operations in his mind. Nothing is more 
desirable for an Entry Clerk or Book-keeper than to 
have a thorough knowledge of the aliquot parts of 
100 or 1000. Many remunerative situations have 
been obtained by those who thoroughly understood 
the practice, though their general education was very 
limited. Constant practice will enable anyone to 
give the products as fast as the questions are given, 


Another mode of multiplying, when the multiplier 
can be divided into factors, is an improvement on 
the common method ; but to multiply in a single line 
is still better. ‘The objection to this method is that 
when an error occurs in the first line it-will run into 
the second. 

EXAMPLE: — 

1234 

124 

4936 
14808 


153016 


Here we multiply through by 4; now, since 12 is 
3 times 4, if we tuultiply the first line by 3 we have 
the product of 12 ina single line. Quite a variety 
of examples can be worked in this way. 


HOW TO PROVE MULTIPLICATION BY CASTING 
OUT THE NINES. 

RULE.— Find out the excess over nine in your multipli- 
cand and multiplier, and if the excess in the product of 
these excesses is the same as the excess in the product, 
the operation is correct. 


EXAMPLE :— 
oo 4 
| Leap ae 8 


a ere 
~] 
“xX x 
= 


153016 


MULTIPLICATION 


——— BY —— 
SOUARING NUMBERS. 
= 1 spans 
9? 4 13? — 169 
S7=. 9 142 — 196 
4? .. 16 15? = 225 
2 25 167 — 256 
6? = 36 172 = 289 
7? == 49 , 18? = 324 
8? — 64 19? — 361 
9? = 81 20? = 400 
10? —100 212 = 441 
11? —=171 222 — 484 


Notrre.—The product of any two numbers is equal to the 
square of the mean, minus the square of half their differ- 
ence. The meax isa number as much greater than the 
less, as it is less than the gréater. 

RuLeE.—From the square of the mean subtract the square 
of the difference between either of the given numbers and 
the mean. 


EXAMPLE 1. 


17 
13 15* — 4 o= 221 


221 


REMARK.— 15 is as much greater than 13 as :t is less 
than 17, 15 is therefore the mean between 17 and 13. 
Half the difference is 2, the square of 2 is 4; or, we may 
say, the difference between the mean and either of the giv- 
en numbers is 2, square of which is 4. 


49 


EXAMPLE 2.---What cost 19 books at 13 cts. each ? 


SoLuTIon.—The mean of 13 and 19 is 16, the square of 
which is 256 —9, the square of a/f the difference of the 
given numbers = 245. 


EXAMPLE 3.—What cost 13 tons of hay at $14.00 
a ton? 
Thus: 13? + 13 = 182 
Where the difference of two numbers is a unit, we 
add the less number to its square, for 
13 X 13 = 14 times 13. 


EXAMPLE 4.—What cost 16 ounces of gold dust 
at $17.00 an ounce? 


Thus : 167 +. 16 = 272. 
Table of Square Numbers continued. 
26? — 676 
277 = 729 
237'== 529 287 = 784 
247 — 576 29? = 841 
257 == 625 30? = 900 


Here let the student observe that the two right- 
hand figures in the square of any number, as much 
less than 25 as another is greater, are the same in 
one case asin the other. For example, in the squares 
of 23 and 27, the one as much less as the other is 
greater than 25, the two right-hand figures are the 
same. ‘This law holds true in all cases. 


EXAMPLE 1.—What cost 23 shad at 27 cts. each ? 
DA Ven A ie OTL 


EXAMPLE 2.—What cost 26 tons of hay at $27.00 


a ton? 
26? + 26 — 703 


50 


Table of Square Numbers continued. 


317 = 961 197 == (361 
CUE Lae 18? = 324 
33° ==) 1089 LZ? c=) 1289 
347 = 1156 16% 25256 
$57 == 1225 157)== 1225 
SO =F 12960 147° i396 
377 == 1369 13? — 169 
387 = 1444 12? = 144 
Sones Lee 1H poet Brg 
40? — 1600 10? == 7100 


The first column is placed here to afford the stu- 
dent an opportunity of observing that the ‘wo right- 
hand figures in the square of any number which is as 
much less than 25 as the other is greater, are the 
same in the former case as in the latter; or, which 
is the same thing, the two right-hand figures in the 
square of any number as much és than 20 as anoth- 
er is greater than 80 are the same in one case as in 
the other. 


Table of Squares continued. 


4}? — 1681, 9? 46? = 2116, 4? 
42? — 1764, 8? 47? == 2209, 3? 
43? — 1849, 7? 48? — 2304, 2? 
442 — 1936, 6 49? — 2401, 1° 
45? — 2025, 5? 50? = 2500, 0° 


The student need have no difficulty in remember- 
ing the squares in the above columns. Observe that 
the two right-hand figures in the square are in every 
instance the square of the difference between the — 
units figure of the root and 10; the two left-hand fig- 
ures in the square will also be easily remembered if 
we observe that the number is formed by adding 1 
less than the right-hand or units figure of the root 


61 
to the square of the left-hand, or tens figure; thus, 
in the square of 47 the left-hand figures are 22, or 1 
less than 7 added to the square of 4, thus: 

47 == 16 + (7—1) =22. 

The right-hand fignres are obtained by subtracting 
7 from 10 = 3; the square of 3 = 9 

Note.— When the square of the difference between right- 
hand figures and ten is not over 9, prefix a cipher, as in 
above case. 

EXAMPLE 2.—437 = 10 — 3= 7 = 49, right- 
hand figures; 4 X 4 +- 2 = 18, left-hand figures ; 
therefore 43° — 1849. 


Table of Squares continued. 


Olt o01 567 — 3186 
ra) 2704 WY feditomensts PY 4!) 
Dose 2509 58? == 3364 
54? — 2916 TE bipeeegitey 23H) 
Wes OULD 60? == 3600 


This square can also easily be remembered if we 
observe that the left-hand figures in every instance 
may be produced by adding the right-hand figure of 
the root to the square of the left-hand figure, and the 
number expressed by the two right-hana figures in 
the square, is the square of the right-hand figures in 
the root. . 

Example, square 56. Here we say 6? = 36, which 
are the right-hand figures in the square. Again, 
5? + 6 = 31, which are the left hand figures in the 
square. 


Table of Squares continued. 


Beemer ete. te P10) OF ee AQ 00 4* — 576 
ee Pte 4A ee O41 2 5* == GBD 
age OO ath = 1692) 72% O 184.) 967 == 676 
Bae OG nL 4 1905 8 fos Doeo ae 27° => 729 
Cie oa hye 229) 14> 47 G., 28) C84 


52 


16? — 256 


66 —= 43856 7h" = D620). oo ee 

677 — 4489. 177=- 289 | 767 5776 © 807 = 900 

687 — 4624) 3/188 = 324 7 77 5929) ae 

69274761) 197 Sols 006 eee 

707 — "4900. © 207 == 4000 70? = 6247 25s 
80? = 6400 


In the above tables the columns of squares on the 
right are thus placed with their roots, in order that 
the student may associate them with the squares of 
the left-hand column. He will of course observe 
that two figures (units and tens ) are in regular suc- 
cession, the same in one column as in the other. In 
the following columns the same principle will be 


observed. 


1 Gob Lalo 1361 Oi4=2"8281 2 9a0 sed 
S20 C7240 9 1 By 324 927 -2"S464 28ers 
So 0909 | 1d) ag 937 S649 aia 
47 (0560-16 26 947 — 8836 67== 386 
Fait 1 ya Sea dh meme 43) 957 == 19025 Wap are 
Bie (odOea Laci 962 02 [Ger eee 
Bie OUTU Mla RD O74 9409 ar ee 
Sore 71744 12h eas 95729604) 632. one 
(op eteess TAPMEL I OT dean NIB 9073/9 OO La cane 
90781002 107 == 00) 1 OUt = 0000 ame 


The square of last column may be obtained by 
adding twice the right-hand figure of the root to 80. 


thus: 


917—2X1—=2+80—82. 


EXAMPLE 2,— 99? = 80 + 18 = 98, left-hand fig- 
ures in the square. 


TO SQUARE ANY NUMBER ENDING IN 5. 


RuLE.— Multiply the part preceding the units by 
itself, increase by a unit and prefix the product to 
25, thus: 65?== 6 X 7 == 42 ; prefix 25.== 4225. 


EXAMPLE 2,—72?== 7X 8= 56; 


5620. 


prelxm2ome— 


DIVISION. 


' The work in division can be abbreviated by not 
writing out the product figures, and finding the re- 
mainders as we pass along. 

RULE. — Subtract each product figure as it is formed 
(that is, the right-hand figure of the product ), and when 
it is greater than the figure from which you subtract, carry 
one more to the next product figure than you would other- 
wise carry. 


EXAMPLE :— 
29)15341(529 
8 


2 6 


Say5X9= 45. 5 from 13 = 8; then 5 X24 
4+-1=—15. Our next quotient figure is 2; now 
say 2X9=—18; 8 from 14=—6; 2X2+1-+1=6, 
subtracted from 8 leaves 2, or 26, our next, etc. 

Vhe Italian method of dividing is to place the di- 
visor to the right of the dividend, and the quotient 
underneath it. The advantage in this is, you bring 
the operation closer together and can readily multi- 
ply the quotient by the divisor, to prove the work. 


EXAMPLE :— 


04 


CONTRACTIONS IN DIVISION. 


The table of Aliquot Parts on page 25 can be ap- 
plied to division. 

To divide by 25 multiply the dividend by 4 and 
point off two figures to the right-hand of the product 
as so many hundredths, or take one-fourth of the 
two right-hand figures of the product as so many 
twenty-fifths. 

ExaMPLE 1.— Divide 3757 by 25. 


De ok 
4 


15028 or, 1507, 


EXAMPLE 2.— Divide 437924 by 125. 


437924 
8 


3503392 = 350349 
This method will hold good through the entire ta- 
ble of Aliquot Parts. It is simply the reverse of 
multiplication. Another convenient mode of divid. 
ing is to reduce the divisor into factors. 
ExampLe.— Divide 34969 by 24. 
Dd rece Ok 
6)34569 
4) 576 1— 3 over. 
1 44 0 — 1 over. 


6 X 1+ 3 = 9 true remainder. 


or, 4) 34569 
he aan 
1440—2 
2X 4 + 1 = 9 true remainder. 


FRACTIONS. 


A Proper Fraction is one whose value is less than a 
unit, as 4, #. 
An Jmproper Fraction is one whose value is equal to or 


more than a unit, as 3, 3. 


A Mixed Fraction consists of a whole number and a 
fraction, as 24, 44. 


1.—Multiplying the numerator of a fraction by any 
number multiplies the value of the fraction by that 
number. | 

2.—Dividing the numerator of a fraction by any 
number divides the value of the fraction by that 
number. 

3.—Multiplying the denominator of a fraction by 
any number divides the value of the fraction by that 
number. 

4.—Dividing the denominator of a fraction by any 
number multiplies the value of the fraction by that 
number. 

5.—Multiplying both numerator and denominator 
of a fraction by any number does not change the 
value of the fraction. 

6.—Dividing both numerator and denominator of 
a fraction by any number does not change the value 
of the fraction. 


56 


To Higher Terms 
How many fourths in $? 


SoLuTion. —In 1 there are }, and in 4 there are $ of 3, 


Ores ' 
How many sixths in ae 4? 8? §? 
How many eighths in Ae AP EP ee 
How many tenths in a? 4? 3? SP 
How many twelfths in 4? 42? 4? 
How many fourteenths in $? #? 4? #2? 
How many fifteenths in 3? $2? 4? 3? 
How many sixteenths in 4? #? 2? 3? 
How many eighteenths in 3? 4? #3? 8? 
How many twentieths in 4? 3? 73? #3? 


Te Lower Terms. 
How many thirds are equal to ¢? 


SoLUTION.—+# is equal to 2, therefore 4 of the number of 
sixths equals the number of thirds; 4 of 4 is 2. 


How many halvesin 2? $$? 8? 149? 
How many thirdsin #2? ¢? §? 48? 
How many fourths in $2? 8? 3%? le 
How many sixthsin 18? ? 8? 48%? 
How many eighths in 4? 42? 8? 32? 
How many fifths in go? 8h S22? 38? 
How many sevenths in +9? 8? 3%? 32? 
How many ninthsin 427 48? 35? 38? 
How many tenthsin 48? 21? 24? 25? 


To a Common Denominator. 


When fractions have the same denominator, they are 
said to have a Common Denominator. 

Ex.— Reduce % and # to a common denominator. 

SoLUTION.—A common eae ecnage for 3ds and 4ths is 
I2ths; in 1 there are 13s and in # there are # of +4, or 53 
and in § there are % of 14, or 75. 

Reduce 4 and } to a common denominator. 

Reduce 4 and to a common denominator. 


Addition of Fractions. 


What is the sum of 3 and #? 


SoLuTIon.—# equals 38;, and $ equals 3;; 38 plus #5 are 


1f, which equals 135. 


What is the sum 


Of 4 and }? 
Of 2 and 3? 


Of? and st 
Of 3 and 4? 


Of # and #? 

Of 34 and 44? 
Of 34 and 22? 
Of 43 and 5}? 
Of 73 and 83? 
Of 4, 4, and } 


Of 24 and 34? 
Of 2% and 13? 
Of 62 and 52? 
Of 61 and 54? 
Of 4, 4, and 2? 
? Of 4, 4, and 4? 


Subtraction of Fractions. 


What is the difference between 


SOLUTION. — } is equal to 5%, and 


3 2 
# and 8? 
2 

3 


minus ,5, equals 4. 


% from 4? 
# from 8? 
4 from 3? 
epost 
8 from §? 
24 from 3}? 


Subtract 

2from ¢? 3 from 2? 
4 from }? 1 from 3? 
2 from ¥? 4 from 2? 
4 from 3? 2 from #? 


24 from 34? 
31 from 5}? 


31 from 44? 


Multiplication of Fractions, 


How many are 4 times 2? 


SoLuTion.—4 times ~ are 12, which equals 3 or J, 


How many are 3 times 
How many are 4 times 


How many are 7 times 
How many are 4 times 
How many are 8 times 
How many are 8 times 
How many are 5 times 


2? 3 times 2? 
§? 5 times +? 
7? 38 times 2? 
tf? 6 times 7? 
St tunes yee 
2? 8 times 3? 
sco: meses? 


57 


is equal to &; *% 


58 


Division of Fractions. 


How many times is % contained in 4? 


SoLuUTION.—1 contained in 4, 4 times; and if 1 is cone 
tained in 4, 4 times, 4 is contained in 4, 3 times 4 times, 
which are 12 times, and 2 thirds is contained in 4 4 of 12 


times, or 6 times. 


How many times is 2 contained in 2? 
4 

How many times is ? contained in 8? 
5 

itow many times is ? contained in 2? 

How many times is 3 contained in 5? 

How many times is 2 contained in 4? 

How many times is § contained in 2? 
8 

How many times is ? contained in 3? 


SOLUTION.—? is equal to $$, and 3 is equal to 42; 439 is 
contained as many times in $2? as 10 is contained 12, which 


is 12 or £ times. 
How many times is ? contained in 2? 
How many times is ? contained in 3? 
How many times is 2 contained in }? 
How many times #is $3? 3? #2? 
How many times 2 is}? $? %? 


Relation of Fractions. 


What part of 2 is #? 


SoLuTION. — 1 is 4 of 2; and, if 1 is 4 of 2, 4 is 4 of 4, 


which is 4 of 2, and # is 8 times 3, or 2 of 2. 
What part of 3 is #? Of 2 is 
What part of 4 is 2? Of 5 is 
What part of 4 is ¢? Of 7is 2: 
What part of 9is 3? OF 5 is 4 of 2? 


sokn eco ICO 
wv wv 


In 3? 
In 5? 
In 4? 
In 7? 
In 5? 
In 4? 


What part of 6 is of 2? Of 7 is 2 of 8? 


What part of % is #? 


SoLuTIon,—} is } of 3, and 3, or ome, is 3 times 4, or 3 of 
3. Since ove is $ of 3, 4 is t of $, which is }%) of %, and # 


is 4 times ;*;, which are +4, or § of 3. 


What. part of #'isi2?,. OL 2 ister), Of fis 22 
What part of $ is $#? Of 2? is #? Of 3 is 23? 


kieducing to Fractions. 


What is 4 of 4? 
So.tuTion.—} of 1 is 3, and if } of Lis }, 4 of 4 is 4 times 
3}, which are }. 


What is 
4 of 5? 4 of 6? tof 7? 4 of 9? 
4 of 5? Z4of 10? 4Fof122 ~4 of 20? 
py of 24? qs of 32? 2 of 6? # of 10? 


What is 3 of §? 


SoLuTIon. — + 


times 3, which are { or $. 


of § is 2. and if } of $18 2, ¢ of § are 2 


What is 
3 1 2 vi 3 6 2 4 1 
% of 13? 2 of $? 8 of §: % of 38? 
3 15 2 1 9 
§ of +3? Zof té? gof wy? 


What is 4 of 4? 


SoLuTion.—4 is one of the 4 equal parts into which a 
unit may be divided ; if we divide each fourth into 3 equal 
parts, each part is + of 4, and since there are 4 times 3, or 
12 parts in all, each part is =); of a unit? 


ANOTHER SOLUTION.—}# equals #,,.and } of ,%, is +4. 


This 


is a simpler solution, but not so explanatory. 


What is 3 of 4? 
What is 4 of }? 
What is 4 of 3? 


What is } 
What is 
What is 
What is 


What is 


of 4? 
of 4? 
of Jy? 
of py? 


of #? 


(Xoo o> eras Uo 


4times 4, which are ,4, and # of # are 2 times ,4, 


What is 4 of 3? 
What is 4 of 3? 
What is } 
What is 3 
What is 2 


4 of 4? 4 of f? 
4 of 4? 4 of 3? 
1 of 3? 4 of +? 
+ of +? % of +? 
1 of 3? 4 of 3? 
4 of 3? lof p>? 
po Of pr? yy of ry? 
SoLurion. — 4 of | is 4, and if 4 of 4 is A, $ of § is 
or &. 
4 of 3? } of 4? 
Zof 8? 40f 8? 
Zof%? sof 4? 
4rnek 
# of 3? € of $? 
Bofe? 3% of 3? 


CONTRACTIONS 


Hal An@ le EG Nis 


To square any number containing 4, as 74, 84. 


RuLE.—Multiply the whole number by the next higher 
whole number, and annex to the product. 


Ex. 1. What is the square of 74? Ans. 564. 
We simply say, 7 times 8 are 56, to which we add 4. 
What will 94 lbs. beef cost at 94 cts. a lb. ? 
What will 125 yds. tape cost at 124 cts. a yd.? 
What will 53 lbs. nails cost at 54 cts. a Ib.? 
What will 114 yds. tape cost at 114 cts. a yd.? 
What will 195 bu. bran cost at 193 cts. a bu.? 


cat ner 


Reason.—We multiply the whole number by the 
next higher whole number, because half of any number 
taken twice and added to its square is the same as 
to multiply the given number by ONE more than it- 
self. The same principle will multiply any two “Ze 
numbers together, when the sum of the fractions is 
ONE, as 84 by 88, or 112 by 118, etc. It is obvious 


61 
CONTRACTIONS IN FRACTIONS. 


that to multiply any number by any two fractions 
whose sum is ONE, that the sum of the products must 
be the original number, and adding the number to its 
square is simply to multiply it by onE more than it- 
self ; for instance, to multiply 74 by 73, we simply 
say, 7 times 8 are 56, and then, to complete the multi- 
plication, we add, of course, the product of the frac- 
tions (} times 4 are ,3,), making 56%, the answer. 


Multiplication, 
WHERE THE SUM OF FRACTIONS IS ONE. 


To multiply any two like numbers together when 
the sum of the fractions is ONE. 

Ro.e.—Multiply the whole number by the next higher 
whole number ; after which, add the product of the fractions. 

Multiply 32 by 34 in a single line. 

Multiply 4 < #, which gives ~,andsetdown 3% 
the result ; then we multiply the 3 in the multi- 34 
plicand, increased by unity, by the 3 in the 
multiplier, 3 X 4, which gives 12 and completes 12,5, 
the product. 

Multiply 72 by 72 in a single line. 

Multiply 2 X 2, which gives 5%, andset down 7 
the result ; then we multiply the 7 in the multi- 73 
plicand, increased by unity, by the 7 in the 
multiplier, 7 X 8, which gives 56 and completes 563%; 
the product. 

Multiply 114 by 112 in a single line. 

Multiply 3X 4, which gives 3, and set down 114 
the result; then we multiply the 11 in the 113% 
multiplicand, increased by unity, by the 11 in —— 
the multiplier, 11 X12, which gives 132, and 1323 
completes the product. 


62 
CONTRACTIONS IN FRACTIONS. 


To multiply any two numbers together, each of 
which involves the fraction 4, as 74 by 94, etc. 

RuLe.—To the product of the whole numbers add half 
their sum plus 4. 

1, What will 34 doz. eggs cost at 74 cts. a doz.? 

Here the sum of 7 and 3 is 10, and half this 34 


sum is 5, so we simply say, 7 times 3 are 21 73 
and 5 are 26, to which we add 4. 264 


N. B. 1f the sum be an odd number, call it one less to 
make it even, and in such cases the fraction must be 3. 

2. What will 114 lbs. cheese cost at 94 cts. a lb.? 

3. What will 84 yds. tape cost at 154 cts. a yd.? 

4. What will 74 lbs. rice cost at 134 cts. a lb.? 

5, What will 1U§ bu. coal cost at 124 cts. a bu.? 

ReEason.—In explaining the above rule, we add 
half their sum, because half of either number added 
to half the other would be half their sum, and we 
add 4, because 4 by 4 is 4. The same principle will 
multiply any two numbers together, each of which 
has the same fraction ; for instance, if the fraction 
was 4, we would add one-third their sum; if #, we . 
would add three-fourths their sum, etc. ; and then, to 
complete the multiplication, we would add, of course, 
_ the product of the fractions. 


MULTIPLYING ANY TWO NUMBERS TOGETHER, EACH 
INVOLVING THE SAME FRACTION. 

RuLe.-—To the product of the whole numbers, add the 
product of their sum by either fraction; after which, add 
the product of their fractions. 

1. What will 113 lbs. rice cost at 92 cts. a lb.? 

Here the sum of 9 and 11 is 20, and three- 113 
fourths of this sum is 15, so we simply say,9 93 
times 11 are 99 and 15 are 114, to which we 
add the product of the fractions (9). 114,5, 


63 


CONTRACTIONS IN FRACTIONS, 


What will 72 doz. eggs cost at 8% cts. a doz. ? 
What will 62 bu. coal cost at 62 cts. a bu. ? 
What will 452 bu. seed cost at 32 dols. a bu.? 
What will 32 yds. cloth cost at 53 dols. a yd.? 
What will 172 ft. boards cost at 132 cts. a ft.? 

7. What will 182 lbs. butter cost at 182 cts. a lb.? 

N.B. If the product of the sum by either fraction 
is a whole number with a fraction, it is better to re- 
serve the fraction until we are through with the whole 
numbers, and then add it to the Lieec of the frac- 
tions ; for instance, to multiply 34 by 74, we find the 
sum of 7 and 3, which is 10, and one-fourth of this 
sum is 2}; setting the 4 down in some waste spot, 
we simply say, 7 times 3 are 21 and 2 are 23; then, 
adding the 4 to the product of the fractions (9), 
gives 78, making 23,9, Ans. 


LEA ara ae 


MULTIPLYING ANY MIXED NUMBERS. 


Ru Le 1.—Multiply the whole numbers together. 
2.—Multiply the upper digit by the lower fraction. 
3.—Multiply the lower digit by the upper fraction. 
4.—Multiply the fractions together. 
5.—Add these four products together. 


122 
ExaMPLE.— Multiply 122% by 9%. 93 
1.—We multiply the whole numbers = 108 
2.—Multiply 12 by i — 9 
3 —Multiply 9 by $ = 6 
4.—Multiply # by $ = 0-8; 


5.—Add these four aie together, 123.5 
and we have the complete result. 

N.B. When the student has become familiar with 
the above process it is better to do the intermediate 
work mentally, and, instead of writing out the partial 
products, add them in the mind as you pass along, 
and thus proceed very rapidly. 


64 


CONTRACTIONS IN FRACTIONS. 


PRACTICAL BUSINESS METHOD 


FOR MULTIPLYING ALL MIXED NUMBERS. 


Business men generally. in multiplying the mixed 
numbers, only care about having the answer correct 
to the nearest cent; that is, they disregard the frac- 
tion. When it is a half cent or more, they call it 
another cent; if less than half a cent, they drop it. 
And the object of the following rule is to show the 
easiest and most rapid process of finding the product 
to the nearest unit of any two numbers, one or both 
of which involves a fraction. 

Multiply 81 by 104. 

Here we simply say 10 times 8 are 80 and} 8} 
of 8 is 2, making 82, and } of 10 is 2, which 104 
makes 84; then 4 times } is 54, making 84,4 
the answer. 84545 


TO MULTIPLY ANY TWO NUMBERS TO THE NEAREST UNIT. 


GENERAL RULE 1.—Multiply the whole number in the 
multiplicand by the fraction in the multiplier to the near- - 
est unit. 

2.—Maultiply the whole number in the multiplier by the 
fraction in the multiplicand to the nearest unit. 

3.—Multiply the whole numbers together and add the 
three products in your mind as you proceed. 

N. B. In actual business the work can generally be 
done mentally; only easy fractions occur in business 

N. B. This rule is so simple and so true, according to 
all business usage, that every accountant should make him- 
self perfectly familiar with its application. There being 
no such thing as a fraction to add in, there is scarcely any 
liability to error or mistake. By no other arithmetical 


process can the result be obtained by so few figures. 


65 


CONTRACTIONS IN FRACTIONS. 


EXAMPLES FOR MENTAL OPERATION. 


EXAMPLE FIRST. 


Multiply 114 by 84 by business method. 114 
Here 4 of 11 to the nearest unit is 8, and4 84 
of 8 to the nearest unit is 3, making 6, so we 
simply say, 8 times 11 are 88 and 6 are 94, Ans. 94 


Reason.—{ of 11 1s nearer 3 than 2, and } of 8 is nearer 
3 than 2. Make the nearest whole number the quotient. 


EXAMPLE SECOND. 


Multiply 72 by 9% by business method. 

Here ? of 7 to the nearest unit is 3, and # 7? 
of 9 to the nearest unitis 7; then 3 plus 7is 92 
10, so we simply say, 9 times 7 are 63 and 10 
are 73, Ans. 73 


EXAMPLE THIRD. 


Multiply 234 by 194 by business method. 

Here 4 of 28 to the nearest unit is 6, and4 234 
of 19 to the nearest unit is 6; then 6 plus 6is 194 
12, so we simply say, 19 times 23 are 437 and — 

12 are 449, Ans. 449 


EXAMPLE FOURTH. 


Multiply 128% by 25 by business method. 128% 
Here 2 of 25 to the nearest unit is 17,so 26 
we simply say, 25 times 128 are 3200 and 17 - 
are 3217, the answer. $217 


PRACTICAL EXAMPLES FOR BUSINESS MEN. 


1 What is the cost of 174 lbs. sugar at 182 cts. 
per lb.? 


66 


CONTRACTIONS IN FRACTIONS. 


Here # of 17 to the nearest unit is 13, and = 174 
4 of 18 is 9, 13 plus 9 is 22, so we simply 1832 
say, 18 times 17 are 306 and 22 are 328, the ——— 
answer. Bo: 26 

2. What is the cost of 11 lbs. 5 oz. of butter at 
334 cts. per Ib. ? 

Here § of 11 to the nearest unitis 4,and 11 - 
1g of 38 to the nearest unit is 10; then 4 334 
plus 10 is 14, so we simply say, 33 times 11 
are 363 and 14 are 377, Ans. 

3. What is the cost of 17 doz. and 9 eggs at 
124 cts. per doz.? 

Here 4 of 17 to the nearest unit is 9,and = 17,5 
fz of 12 is 9; then nine plus 9 is 18,sowe 124 
simply say, 12 times 17 are 204 and 18 are 
222, the answer. 

4, What will be the cost of 153 yds. calico at 
124 cts. per yd.? Ans. $1.97. 


$3.77 


, 


—s 


DECIMALS. 


To reduce a decimal to a common fraction. 


Ru.e.— Write the decimal as it stands, omitting the deci- 
mal point, for the numerator. For the denominator write 
1 with as many ciphers annexed as there are decimal places 


in the numerator. 


EXAMPLES, 


Reduce .25 to an equivalent common fraction. 
Ans. 7°5, which reduced to its lowest terms = 4. 

Reduce .375 to acommon fraction. Ans. 73,4°5 = 2. 

Reduce .1875 to a common fraction. Ans. 4. 


Reduce .625 to a common fraction. Ans. %. 


To reduce common fractions to decimals. 

RuLEe.—Annex ciphers to the numerator, and divide by 
the denominator, prefixing a point to the quotient. There 
must be as many places in the quotient as there have been 


ciphers annexed; if not enough, prefix ciphers. 


68 


EXAMPLES. 


Reduce # to a decimal. 


4)3.00 
.75 = hy Ans. 
Reduce # to a decimal. Ans. .375, 
AI pa HS Ans. .8571-++. 
66 4 73 be bs 
ce 3 66 6é 


MULTIPLICATION OF DECIMALS. 


RuLe.— Multiply as in whole numbers, and from the 
right of the product point off as many figures for decimals 
as there are decimal places in both mult¢plicand and mul- 
tiplier. 

If there are not figures enough in thef product, prefix 
ciphers. 

EXAMPLES. 


Multiply 4.25 by 6.5. 


4.25 
6.5 
2125 
2550 
27.625 Ans. 
Multiply 84.5 by 4. 
pM! LASS SD 
4 
338.0 
Multiply 6.425 by 4.25. 
3 ibe Mg ES Ge 
* LOD UN Oe 
He 25. ** 6.0025 
275 ** 3.0025 


«18.625 « 5.25. 
What is the cost of 12% lbs. at 64 cts. per-lb? 
What is the cost of 72 yds. at 182 cts. per yd? 


NOTE.—It is sometimes more convenient to change common frac- 
tions to decimals before multiplying. 18} x 124 = 18.75 x 12.5. 


69 


UNITED STATES MONEY. 


10 Mills(m)= 1 cent, ct. 
10 Cents =1dime, d. 
10 Dimes = 1 dollar, §. 
10 Dollars = 1 eagle, E. 


The origin of the symbol $, or the United States 
dollar mark, has been ascribed to several sources. 
By some it is supposed to represent the ¢/ written 
upon the S, denoting U. S. (United States). Some 
think it is a modification of the figure 8, having refer- 
ence to 8 reals, or piece of Eight, as the dollar was 
formerly called; others, that it represents the “ Pil- 
lars of Hercules,”’ which were stamped on the Pillar 
Dollar ; and others, still, that it is a combination of 
the initials P. and S., from the Spanish Peso Duro, 
signifying Hard Dollar. As it is used in Portugal to 
note the thousands’ place, it is probable that it ori- 
ginated in that country: a Mil-reis, or thousand. reis, 
is written thus, 1$000. 


The term Dime is from the French dsme, mean- 
ing ten. 


The term Cent is from the Latin centum, a 
hundred. 


The term Mill is from the Latin mz//e, a thousand. 


70 


UNITED STATES COINS, 


The Gold Coins are the Double Eagle, $20.00 ; 
Eagle, $10.00 ; Half Eagle, $5.00; Quarter Eagle, 
$2.50 ; three dollar piece and dollar. 


The Fifty Dollar Piece is not a legal coin. The 
Half Copper Cent is no longer coined. The Mill 
is not a coin. Gold coin contains nine parts gold 
and one part copper and silver. 


The Silver Coins are Dollar, Half Dollar, Quarter 
Dollar, Dime, Half Dime and Three Cent Piece. 
Silver coins contain 9 parts silver and 1 part copper. 

The Nickel Coins are the Cent the new Three 
Cent Piece and new Five Cent Piece. 


The Nickel contains 88 parts copper and 12 parts 
nickel. 


The Copper Coins are the Cent and Two Cent 
Pieces. The Two Cent and Cent Pieces are made 
of nickel and copper. 

One Eagle (Gold) weighs 258 troy grains. 

One Dollar (Silver) weighs 412.5 troy grains. 
One Cent (Copper) weighs 168 troy grains. 
23.2 grains of pure gold = $1.00. 


Gold Coins prior to 1834, like that of England 
= 88.8 per dwt. By an act of Congress of 1834 
its value was made 94.8 cents per dwt. The old 
U.S. Eagle coined previous to 1834 is worth $10.66.8. 

By an act of Congress the payment of debt with 
coin is regulated as follows: 

All Gold Coins at their respective value for any 
amount. The Half Dollar, Quarter Dollar, Dime 
and Half Dime at their respective value for debt 
under $5.00. The Three Cent Piece for debts of 
any amount under thirty cents. The one cent pieces 
for debts of any amount under 10 cents. 


71 


STATE CURRENCY. 


The money of this country before the adoption 
of the decimal currency by Congress in 1786 was in 
the denominations of pounds, shillings, and pence. 
The Colonial notes which were then in circulation had 
depreciated in value; and the number of shillings 
equivalent to a dollar at that time are given in the 
following table :— 


New ENGLAND CURRENCY. 


New England States, Virginia, ' $i 6s 27, 
Kentucky, and Tennessee, Vso) Lbs.cts, 


——— rs 


New YorK CURRENCY. 


New York, Ohio, Michigan, $1°—= 8s, == 962. 
and North Carolina, 15, ==) 124 cts. 


PENNSYLVANIA CURRENCY. 


Pennsylvania, New Jersey, at Sli 26.00. Us 
aware, and Maryland, Ise=13% cts; 


GEORGIA CURRENCY. 


Georgia and South Carolina, $1 = 4s. 8d. = 56d. 
ieee ueeCts. 


72 


The Coins of Foreign Nations, 


With their value in United States coins, as determined 


by the recognized standard at the Mint 
in Philadelphia. 


| Gm pn Pa BD BN a te 2 eS 


COUNTRY. 


$ 

AQBtCTiA wits wie oe wate -|Fourfold ducat ..........| 9 
bs o\4,0/0:8:0'. sna son's (4 HOFIDS (NEW) vino x'ae wares 1 

SeAEe Peeves lettis clots leer eis Duca tires statics sees atatste les Z 
Belgium .....-+-...... 25 TENCE sen ws sew siarseees 4 
lite bales oR eae ics” BO aNtITets sss sce se'ele ase 10 
Central America ..... FPSCRAGE) rd sabe ales o aterbio ee 3 
CET UN Nie dels Toe, A Teal sited sae 6 ose eo 

Chili ....--...2------ 10 pesos (dollar) ........| 9 


Columbia and South 


Value in 
U.S. money 


DENOMINATION. 


America generally..|/Old doubloon ..........- 15 
Columbia ........-... 20 pesos, ‘‘ Bogota” ..... 18 
ay pee e cree recs 20 pesos, ‘: Medellin”’....{19 

, sccesecnnnee 20 pesos, ‘* Popayan” 18 
Costa Rica. ..-+seces. 10) PesOSiesa te roca renee 8 
Denmark © <<. ss e008 AY Crowne 4. 2 we ns ae ee 5 
SET aR ous sctaatete Old ten thaler. Dn eS ort n 7 
Egypt .----ccsccccees Bedidlik (100 piasters)...| 4 
England . ...2seee.ee. Pound, or sovereign (new) 4 
Fraticec ccc ss ss neateO ATA Csiia a enlete vis ete ene 3 
German eas re Cae New 20 marks..........- 4 
Greece .. -cese e120 drachMS o2-scceen-ese 3 
India (British) in Gia tele Mohur. or 15 rupees...--.| 7 
tal. ~ cee oly aie wares QO Tite: h alae choad one cen 3 
JAPAN .--- es ee seceece, 20: VET ~ ceces svccereccwes 19 
Mexito Vy oh sag es a neues Doublponaues ety sate ce 15 
Netherlands.........- 10; Piidera Mun Wares aanlen 3 
New Grenada ........ 10 p: sos (dollars) .....-.. 9 
Norway --++--+++eee-- PU CVO WAS atts s ele, wt wtelae ecets 5 
Perit tegeny aah ea ante AQ-GO1E8. uaa! 19 
SEM ae wie vee eg esiae -|Coroa (crown) «-+++.-... 5 
Russia -..-cccescoees [5 rubles. ..--sccccvcccess 3 
SOE cases + 4's ones ot 100 Tealsrsiv Ah = <tr ae came ee 4 
66 ne ceee cecee cece (80 reals oe ese seee cece cess 3 
SWeden s c«teeecess sss] DUCatmmser wide din tp wh 2 
6 seeececceeeee-(Carolin (10 francs) ..----| 1 
Tunis... .ccccescoee- 2bipiasterswn ety scence es 2 
Turkey wocecccveseesei100 piasters cere ercecce Se ee 


cts. in. 


mV 91 09 © © Ore 


COP ANWWOLOINDW PW ONMAWNHWaATIH DOA QD ORL 


Sli VE He: COLMN Ss. 


—___ ae 


Value in 


U. S. money 


COUNTRY. * DENOMINATION. 
$ 
Peet VOCAL eee ONdirixedOllariss geri ws «se 0 
SE poh tian a.ehk ale >to New florist. « wesepisssslee shee 0 
SOMO alive's; ¢.0 «10,2 New Union dollar....... ) 
Belgium’ .-..+-.+,..... S4Tance +: eee tides eoee tO 
Bolivia ..... Shales ore coats New dollar ......-. wee 
Prasil .is\> o* tetrateare fot Wouble milreis .........-. 0 
Seer aclaee oe sli) eno 120) CONtSec.cvee we coeneese t 0 
Central America...... Dotlars set a ets la Bee. 0 
WETS Hs ca aches be oa dicts Olo Gollarvows « ce sie tioies 0 
66 SAP DOE Date Ee New dollar........- tO 
OTR ES SSeS ae -|Dollar (English mint) ...| 0 
Ree Mr ctarevels siatsto\le «i? ie Oo tateacs Sot kactsaow 0 
Be enmark ice ce s xSialse'ss 2 rigsdaler ..----.+--- +. 1 
Egypt SICH BaAee Piaster (new) Morte vete Oisie es (@) 
England ....... ./Shilling (new) --+-e+---- 0 
a tees -| Shilling Oa ek 0 
66 Se MEL OLiiieecterele sake crete Selene nO 
TANCE: | «1s pet TaATiCS ce caretelee stelerii eleists 0) 
North German States. Thaler, before 1857 ...... 0 
-|Thaler (new) biG AOR TORE ) 
Bath eet States.|Florin «-.. 22. sce. se0- oe 0 
- German ppt se eeee 5 marks (new) -----«---- 0 
Greece. ee ae re aI PCHIIS) we «dies ce a weir 0 
Pendoetan: ca ee Rupee «.... eR etic cys -oi08 0 
TEAL: cc ees esas hs vba BD hivecer assis wee ot Me usieca® 0 
Japan aistaters elateisis erent aise i YEN weer cece cece eeeeee: O 
Care Te La oles bere SOTO att hilerns rath ORO aC oe 0) 
WMIGTICOT. fo ost orien ee Ticats $e fa a ts bo ole 0 
Netherlands.....- 216 gilders .....--....-- 0 
Norway. . --se++-+e/Specie daler..-- .-.+.-./ 1 
New cease heredaue a TSCA ELLE LOD Cisie nvete sts aca’ c) s 0 
Vd REE ie ae rN eer Old:dotlarce: sea e. cereeen 0 
Portugal ......-.--+-- BOO POLS e dis nie ad alavale ete'e,s x be 0 
ROUM ania. ds clench 2 lei (francs), TW widens ahs 0 
Re TION T Agi i is ss ae crewed ere eae Oca ceeaelak 0 
SOAIN (oe mews scces sees 5 pesetas (dollars)... sepean 
SWEET fac. stirs ceciene Riksdaleriwes ti sisiciers eleeie's 0 
SRV IEZOTI AT) Gictisiclec\e« ene DoT AT CSohee henel atone ioretievel slatens 0 
Mbt AS eet a dss ss 0 212"0'= B plasters -.-sseceesesees|Q 


Turkey oeoeereseeseeens 20 piasters ecseeceseescece 0) 


cts. 
95 
45 
68 
9] 
91 
95 
Le 
93 
99 
a1 
99 
09 
03 
03 
21 
20 
42 


HM OOOR ER WOHD + DOHNIBNWDOODAOCOR DON OH AAMAS UHH OPE 


7 


GREAT BRITAIN’S MONEY. 


4 Farthings==1 Penny, d. 
12 pence aE Ing, 5. 
20 shillings ==1 Pound, 4f. 


The Gold coms are the sovereign, which repre- 
sents the pound, and the half-sovereign. The guinea, 
of 21 shillings, and its subdivisions, have not been 
coined since 1816. 


The Szlver coins are crowns of 5s., half-crowns, 
florins of 2s., shillings, the 6¢, the 4d. or groats, and 
3d pieces. 


The Copper coins are the penny, half-penny, and 
farthing, coined at the rate of 24 pence per pound 
avoirdupois 

Bank of England Notes are a legal tender for any 
sum over £9; silver is not a legal tender over 40s. ; 
copper, for not more than 12d. in pennies or half- 
pennies ; or 6d., in farthings. ; 


% is acontraction of “brae, s. of solidi, d. of de- 
narii, and g. of guadrantes ; farthing is another word 
for fourthing. } 

The word sterling is supposed to be derived from 
the first coiners of English silver, who came into 
England from Germany in the reign of Richard L., 
and were called EZasterlings. It is used to distinguish 
the currency of Great Britain from that of the Colo- 
nies, and from some continental money bearing the 


same denominations. © 


75 


Valuable Information for Business Men. 


NOTES. 


Demand Notes are payable on presentation, without 
grace, and bear legal interest after a demand has 
been made, if not so written. An endorser on a de- 
mand note is holden only for a limited time, variable 
in different States. 

A Negotiable Note must be made payable either to 
bearer, or be properly endorsed by the person to 
whose order it is made. If the endorser wishes to 
avoid responsibility, he can endorse “ without re- 
course.” , 

A Foint Note is one signed by two or more persons, 
who can each become liable for the whole amount, 

Three Days’ Grace are allowed on all time notes, 
after the time for payment expires ; if not then paid, 
the endorser, if any, should be Jegally notified to be 
holden. 

Notes Falling Due Sunday, or on a legal holiday, 
must be paid the day previous. 

Notes Dated Sunday are void, 

Altering a Note in any manner, by the holder, makes 
it void. 

Notes Given by Minors are void. 

The Maker of a Note that is lost or stolen is not re- 
leased from payment if the amount and consideration 
can be proven. 

Notes Obtained by Fraud, or given by intoxicated 
persons, cannot be collected. 


An Endorser has a right of action against all whose 
names were previously on a note endorsed by him. 


76 


BILLS OF EXCHANGE, DRAFTS, 
ACCEPTANCES. 


A Lill of Exchange, or Draft is an order drawn by 
one person, or firm, upon another, payable either at 
sight or at a stated future time. 

Lt becomes an “ Acceptance” when the party upon 
whom it is drawn writes across the face ‘‘ Accepted,” 
and signs his name thereto; and is negotiable and 
bankable the same as a note, and is subject to the 
same laws. 

In many States both Sight and Time Drafts are en- 
titled to three days’ grace, the same as notes ; but if 
made in form of bank check, “ pay to,” without the 
words “‘at sight,” it is payable on presentation, with- 
out grace. 


7 


HOW TO ENDORSE A CHECK. 


Very few otherwise intelligent and educated peo- 
ple understand how to properly endorse a Bank 
check payable to their order, and few realize the in- 
convenience they cause, by placing their endorse- 
ment in an awkward position. An observance 
of the following rules will enable anyone to place 
their signature in the proper place. 

1. Write across the back—not lengthwise. 

2. The top of the dace is the 4ft end of the face. 

3. To deposit a check, write ‘‘ For Deposit,” and 
below this your name. A clerk not having power of 
attorney to sign or indorse checks, can deposit his 
firm’s checks by writing on the top of the back ‘‘ For 
deposit only to credit of abet Bs etal 
and below this write his own name. 

4. Simply writing your name on the back of a 
check signifies thatit has passed through your hands, 
and is payable to bearer. 

5. Always indorse a check just as it appears on 
the face. For instance, if the check is payable to 
*¢ G, Read,” indorse “‘G. Read ;” if to “ Geo. Read,” 
indorse *‘Geo. Read ;” if to “George F. Read,” in- 
dorse “George F. Read.” If the spelling of the 
name on the face of the check is wrong, indorse first 
just as the face appears, and below, the proper way. 
For instance, the check is payable on face to “‘ George 
Reade ;” indorse “ George Reade,” and below this 
first indorsement write what it shouid have been, 
“George Read.” 

6. If you wish to make the check payable to some 
particular person, write, “‘ Pay to_[E3&son’s NAmx] 
DROLET st OUR MAME) « 


Note. In England all checks are payable to bearer; but 
in this country all strangers presenting checks for payment 
must be identified by some one known to the bank. 


78 


CALCULATIONS 


USED IN PARTICULAR BRANCHES OF BUSINESS. 


To find the value of tous and hundred-weight with- 
out the use of fractions? 

Rue. — Multiply the number of hundred- weight by 5, 
and annex the product to the tons, as so many hundredths 
of tons; then multiply by the given price per ton, and 
point off two decimals. 


EXAMPLES. 


1. What is the cost of 18 tons, 17 cwt. coal at $4 
per ton? 

L2aA 100 18.85 X 4 = 75.40. Ans. $75.40. 

2. What is the cost of 35 tons, 15 cwt. hay at $12 
per ton? 


3. What is the cost of 48 tons, 17 cwt. coal at 
$6.50 per ton? 


To find the value of shillings and pence in the 
decimals of a pound sterling. 


Rou.te.—Multiply the shillings by 5, and call the product 
hundredths. 

Multiply the pence by 4%. and call the product thou- 
sandths. 

The sum of these two values will be the decimal required. 


EXAMPLES, 


1. Reduce 12s. 67. to the decimal of a pound. 
12 iKCD re OU 
6 X 44 = .025 


625 


79 


SPECIAL CONTRACTIONS. 


2. Reduce £187 138s. 37. to the decimal of a pound. 
Laks se) 
3X41 — 0125 
6625 Ans. £187.6625. 


To change aunes to yards. 


NOTE.—An aune is a French measure, equal to 114 yards. 


RuLr.—Annex a cipher and divide by 8. 


EXAMPLES. 


1, In 484 aunes, how many yards? 
4540-275 == 605-02 A ns: 

2. In 3848 aunes, how many yards? In 1265? 
In 1847? 

NoTre.—14 = 3, or1,.. This rule can easily be applied 
to numerous other calculations. The contents of boards 
14 inches thick, etc., may be computed in this manner; 
the selling price of goods in order to gain 25 per cent on 
the cost, and others. 

3. What is the selling price of goods, which cost 
64 cents per yard, to gain 25 per cent? 


640 — 8 = 80 64 
Lo 2o wf of 64. 
80 


To find how many gallons of linseed oil in a given 
number of pounds, at 74 lbs. per gallon. 


RuLe.—Add one-third of the number of pounds to itself, 
and point off one decimal. 


EXAMPLES. 
1. How many gallons in 675 lbs. ? 
675 
225 = 4 of 675 
90.0 Ans. 90 gals. 


2. In 1846 lbs. how many gallons? In 675? 
In 338 lbs. ? 


80 


Another Method 
To reduce pounds, shillings and pence 


to the decimal of a pound. 


Rute. -- Write one-half of the greatest even number of 
shillings as tenths, and if there be an odd shilling write 
five hundredths; reduce the pence and farthings to farth- 
ings, and write their numberasthousandths. If the num- 
ber of farthings is between 12 and 36, add one to the 
thousandths;.1f between 36 and 48, add 2 to the thousandths. 


EXAMPLE :— 
£3 14s. 6 = £38.725. 


Put down the 43, then divide the 14 by 2 and put 
down 7, then multiply 6 by 4 and add 1 to get the 
25. Again, 


£4 15s. 10d. = £4- 732 = £4.792. 


In this example one-half of 15 is 74; therefore 
we put it down in the decimal form .75, and 4 times 
10 are 40, add 2 and we have 42, which added to 
.75 gives .792. 


EXERCISES. 


£2 '88.-6d." 6. 43S 8Ss20e Tea 2s ee 
216468. Oda 0-7 Genes, Od wl 2g) os alos 
: £8 10s. 3d.y 8. h4. Tsi 4d. odd 2S Teed 
£9 4s. 5d. 9. £6 9s.5d. 14. £6 18s. 8d. 
. £1025. 8d. 10. $1 11s. 6d. 15. £5 16s. 73d. 


Cr yw OF DO = 


To reduce pounds to dollars. 
Rutle.-- Multiply the number of pounds by $4.84. 
EXAMPLE : — 

Reduce £3.725 to dollars a cents. 
3.725 X 4.84 = 18.029. 


81 


LNG IO Ree) he 


Interest is the money which is paid for the use of 
money. 

The Principal is the sum for the use of which interest 
is paid. 

The Rute is the per cent. of the principal paid for an 
given time. ; 

Norte. — When no time is mentioned, ser annum, or by the year, is 
understood. 

The .fmouwnt is the sum of the principal and interest. 


Simple Interest is the interest on the sum loaned 
for the given time, at the given rate. 


Legal Interest is the interest according to a certain 
rate per annum, fixed by law. 


Nore 1. —A higher rate of interest than that prescribed by law is 
termed wsury, and is prohibited by law. 


Nore 2.— \When the rate per cent. is not named in notes, or other 
business documents, the legal rate must be taken. 


NoTE 3.—In most of the States, and on debts due in the United 
States, 6 per cent. is the legal rate, although a higher rate may be 
agreed upon by special contract. 

When no rate is mentioned, the legal rate is understood. 
Interest may be simple or compound. In simple interest 
the principal alone draws interest. Interest on interest 
remaining unpaid is considered illegal. In compound in- 
terest the entire amount due at regular intervals is con- 
verted into a new principal. 1t is compounded annually, 
semi-annually or quarterly, and sometimes monthly, ac- 
cording to agreement. 

To be an expert in computing interest it is necessary to 
be fami‘iar with all the methods, and apply the one that 
will give the correct result with the least labor. To com- 
pute interest for years, multiply the principal by the per 
cent., and you have the interest for one year. 

EXAMPLE 1. — What is the interest on $84 at 6 per cent. 


for 3 years? 
$84 X .06 = $5.04 X 3 = $15.12. 


82 


INTEREST. 


IF THE TIME CONSISTS OF MONTHS. 


Multiply the principal by the per cent, and you have 
the interest for 1 year. 1 month being ;; of a year, 
divide by 12 for the interest of 1 month; multiply 
this result by the number of months for the interest. 


EXAMPLE 2.—What is the interest on $120, at 8 per 
cent for 8 months? 


ComMMON METHOD. CANCELLATION. 
120 
12 ) 9.60 120 
‘ : 
6.40 


Nots.—If the time consists of years and months, reduce 
the years to months, adding the number of months, and 
proceed as above. 


IF THE TIME CONSISTS OF YEARS, MONTHS AND DAYS. 


Reduce years to months, adding the number of 
months, then place 4 of the number of days to the 
right of the months and proceed as before. 


REMARK. — Placing 4 of the days to the right, is 
reducing the days to the decimal of a month. The 
reason of this is obvious from the fact that we calcu- 
late 30 days for 1 month, then 1 day is J, of a month, 
and 3 days ;3,, or z5, or in decimal form .1; hence 
by taking 4 of the number of days we obtain tenths 
of a month. 


EXAMPLE 3.—What is the interest on $159, at 9 per 
cent, for i year, 4 months, and 12 days: 


83 


INTEREST. 


CoMMON METHOD. 


150 

.09 1 year = 12 months. 
12\.5:50, Nh i 

1125 12 idavs7— tu 

__ 16.4 16.4 months 
18.4500 
By CANCELLATION. 

4 | 150 3X 15 = 45 X 41 = 18.45 
12 g 3 


164 41 41X38 X15 = 18.45 


This operation comes under the head of contrac- 
tions, and can be multiplied mentally. 


ON ALL NOTES THAT BEAR $12 PER ANNUM, OR ANY 
ALIQUOT PART OR MULTIPLE OF 12, 


Any principal that bears $12 per year will bring 
$1 per month; hence, the time in months must be 
the interest. 

ILLUSTRATION. — If the interest for 1 month is $1, 
for 15.3 months it is 15.3 times $1, or $15.30; since 
the multiplication by the figure 1 is altogether super- 
fluous we can dispense with it, and at once say $15.3 
or $15.80. Hence the propriety of the following: 

RULE. — Reduce years to months, add in the given 
months, place § the number of days to the right, and we 
have the interest in dimes. 

EXAMPLE 1.— Required the interest for $150, at 
8 per cent, for 1 year, 6 months, and 11 days, 

150 4 of 11 days = 38. 

08° Years. Months. Days. 

12.00 1 6 11 = 18.38, 

therefore $18.33 dimes, or $18.36. Ans. 


84 


INTEREST. 


A note that bears an aliquot part or multiple of 
12; such as 6, 18, 24, 36, etc. 


EXAMPLE 2,—What is the interest on $300, at 6 per 
cent, for 3 years, 6 months, and 18 days? 


$300 4, 0f 13-days’—— 6, 
.06 Years. Months. Days. 
18.00 3 6 Loa 2: 
$42.60 Interest at $12 a year. 
21.30 i “$6, or 4 of $12, a year. 
63.90 
CANCELLATION. 
300 
12) hohe 213 x 300 = 63.90 
( 


Note. — To find the time any sum will double itself at 
simple interest, simply divide 100 by the rate per cent. 


ANOTHER METHOD TO COMPUTE INTEREST FOR DAYS. 


Rute. — Find the interest for 1 year and divide by 360 
(the number of interest days ina year), and multiply by 
the number of days for the interest. ‘ 


EXAMPLE.—Required the interest on $720, for 60 
days, at 8 per cent. 


$720 
.08 CANCELLATION. 
360 ) 57.60 ( .16 120 2 
360 60 369 | 60 
2160 9.60 | 8 X 60 X 2 = 9.60 
2160 


RemaARK. — In using 360 for a divisor the cipher 
may be rejected, because it avails nothing in dividing, 
and makes the divisor ten times as short ; your answer 
will be mills instead of cents, as before; cut off the 
tight hand figure, and you have the interest in cents. 


4 


85 


INTEREST, 
For 10 per cent add 4. 
ve tt multiply by 2. 


ExAaMPLE.— What is the interest of $124, at 7 per 
cent, for 54 days? 


124 4 | 31 
an 3g | 124 
1.116 | 54 6 
.186 = }. 7X 6X 31 = 1.302. 
$1.302 


Bankers’ Method. 


Banking business being nearly all transacted on 
the basis of 30, 60, and 90 days, the work can be 
very much abbreviated and the interest, sometimes, 
obtained without any calculation whatever. 

The following example will best illustrate this rule. 


ExAaMPLE.—What is the interest on $120, for 60 
days, at 6,per cent? 
120 
3B | 60 10X 120 =1.200 Ans. 
8 


Observe that in this case we cancel the factors in 
the time and rate, and that the figures in the principal 
remain unchanged, therefore: 

Rue. — For any note at 6 per cent for 60 days remove 
the decimal point two places to the left, and you have the 
interest. 

EXAMPLE.—Required the interest on $350, at 6 per 
cent, for 60 days. 


SOLUTION. — Remove the point in the $350 two 
places to the left, thus: 3.50, and you have the result. 


86 


INTEREST. 


This is the shortest and best rule for days that can 
be adopted, because it may be applied in any per 
cent and any number of days. The great superiority 
of this rule consists in its simplicity, and when once 
understood is not readily forgotten. 


Some accountants have a different rule for every per 
cent, many of which are not only apt to be forgotten, 
except by those who apply them daily, but are actu- 
ally deduced from the above rule. 


In reckoning 360 days instead of 365 gives gz, or 
7x, too much. But the difference is so small that in 
ordinary transactions it is not noticed. It is now 
universally adopted in all business transactions. 

To find the accurate interest divide by 365 instead 
of 360, 7 

When the time is less than 1 month, the cents in 
the principal may be disregarded, because the inter- 
est on ¢hat sum for ¢ha¢ time would not amount to a 
cent; when less than 2 months, all under 50 cents, 
when less than 3 months, all under 33, and so on. 

To illustrate we will give the table of divisors for 
the different per cents. Any sum multiplied by the 
time in days, and divided by the number opposite the 
per cent, will give the interest at that per cent. 


At 5 % divide by 72 At 12% divide by 30 


“6G yp ‘6 73 60 (a3 15 % be ‘é 94 
6“ 7, % 66 66 52 6“ 20 We 66 66 18 
73 8 ve 6 66 45 6c 24 70 6b 66 15 
be “6 6 (74 6 6é 

2H 40 A OA esate 
66 10 % 66 66 36 


It will be observed that these divisors are obtained 
by dividing 360 by the rate per cent ; and the student 
will have to retain in his memory a different divisor 
for every per cent when, by using 36, once for always 


87 
INTEREST, 


he need remember but one. The great advantage of 
using 386 must at once be admitted. Some authors 
use 12 and 30 which, of course, is the same thing ; 
we will now give a few solutions of problems as solved 
by the old method, and also by cancellation, that the 
student may perfectly understand them. 
EXAMPLE 1.—What is the interest of $80 for 1 year, 
° months, and 12 days? 
80 1 yr., 9 mo., 12 days = 522 days. 
.06 
480 
522 
960 
960 
2400 36 
360 ) 250560 ( $6.96 BY) 529 
2160 87 X 80 = 6.960. 
3456 
3240 
2160 
2160 
The student will here notice the vast amount of 
labor saved in the cancelling method. 
EXAMPLE 2.— What is the interest on $48 for 2 
years, 3 months, and 6 days, at 8 per cent? 
2 years, 3 months, 6 days = 816 days. 
6 | Ap 8 


BG 8 
| 816 136 X 8 X 8 = 8.704 


Another short rule for computing interest is called 
THE SIX PER CENT BASIS. 


TO FIND THE INTEREST FOR MONTHS AT 6 PER CENT. 


RuLe.—Multiply the principal by one half the number 
of months; when the principal is dollars only, point off two 


88 


INTEREST. 


places for cents in the product; when dollars and cent 
point off four places. 


EXAMPLE 1.— What is the interest on $153, at 
6 per cent, for eight months? 


sof 8=4 
153 2 1153 x 4 = $6.12 Ans. 
4 12| 6 
wards Chal SS an te $ 4 
$6.12 Ans. 


Solving the above by cancellation will show why half 
the number of months, at 6 per cent, will bring the 
interest. 


TO FIND THE INTEREST FOR DAYS AT 6 PER CENT. 


RuLE.—Multiply by } of the number of days, and the 
product will be the interest in mills. 

ExaMPpLe.—What is the interest on $124, at 6 per 
cent, for 04 days? 


$ of i <2. | 124 X 9 = 1.116. 
a 6| 5A 9 
| 3 


St.i16)) “Ans. 
It will be observed, also, in this that it is an ab- 
breviation of cancellation. 


TO FIND THE INTEREST AT ANY GIVEN RATE. 


RuLE —Find the interest at 6 per cent as above; divide 
by 6 for 1 percent, then multip:y by the given rate; or, 
increase or diminish the result obtained by the rule for 
§ per cent, in the same ratio that the rate is increased or 
diminished. 

For 4 per cent subtract 4. 


~l 
pe) 
Qu. 
Qo. 
OH om obo 


89 
INTEREST. 


When the Time is more or less than 60 Days. 


Increase or diminish in the same ratio as the time 
is increased or diminished. 


For 90 days add 3 itself. 


eta) wee INCI PLY) Vo. 
wou les CL pvicesDye, 
6é 15 “cc be Ae 


“ 45 “ subtract 4. 
Tey Ms eclwide byis. 


ce 10 oe 66 6. 
ica Aceon ft “10. 
SUB Oats “20, 


Norre.—Nearly all business paper is calculated on 30, 60, 
90 days, or an aliquot part or multiple of a month. 


ExAMPLE 1 —What is the interest on $120. for 90 
days, at 6 per cent? 
$1.20 interest for 60 days. 
OUD kee ‘¢ 4 of 60 or 80 days. 


$1.80 interest for 90 days. 


EXAMPLE 2,—What is the interest on $134.24, for 
75 days, at 6 per cent? 


$1.3424 interest for 60 days. 
SOD Dine met Soli, S..,0r + of 60 days: 
$1.6780 
The interest at any other rate can be obtained as 
in preceding rule, or by the following, which will 
show in what time, at the different rates, any number 
of dollars will give the interest in cents corresponding 
with the same figures in the principal. Thus the 
interest on $140 for 90 days, at 4 per cent, is $1.40. 
This rule is no shortér than the other, unless the 
time corresponds with the same figures in the table, 
or when it is an aliquot part of the time in the table. 


90 
INTEREST. 
RuLE. — When the time and rate correspond with the 


time and rate in the table, remove the decimai point twa 
places to the left, as in the preceding rule. 


4 per cent for 90 8 per cent for 45 
5 ce 6é bps 9 ce 66 40 
6 66 “cc 60 10 ce 6< 36 
7 " SoZ 12 %, oa 8) 


The cancelling system is very much preferred to 
this, because it very frequently takes advantage of 
both time and principal, as will be seen in the follow- 
ing solution: 

EXAMPLE. —Required the interest on $540, for 49 
days, at 6 per cent? 

BANKER’S METHOD. 


$5.40 = Interest for 60 days. 


2) 5.40 
2) 2.70 = Interest for 30 days. 
Od Bes ee ede Bhat 
Oh) hua th eee as Ce eh autaes 
a beac nN ty id. LLY 
$4.41 
CANCELLATION. 
g | #49 90 
36 | . x 90 = 4410 


N. B.—Where the time is not an aliquot part of 
the time in the table, but the principal is, reverse the 
operation and point off two places in your time for 
the interest, thus: a note that bears $1.17 in 60 days, 
at 6 per cent, on $117, is the same as $60 for 
117 days. 

NoTE.—We have now conclusively proven that a/ 
the abbreviated processes of computing interest are 
based entirely upon the system of cancellation. 


9t 


INTEREST. 


Indeed, there are many other methods that might 
abbreviate the work, ¢/ you have examples to sutt, 

But the canceling system will give you the advan- 
tage at 44, 6, 8,9, 12, and 15 per cents, and very 
frequently of the time and principal. And, to say 
the least, if the numbers are @// prime, and you can 
cancel zone, you have stated your problem in its 
simplest form to be solved by any other rule. We 
therefore recommend without hesitation the adoption 
of the system of Cancellation, as a general and ui- 
versal rule. The student, by close observation and 
considerable practice, may deduce rules from ¢/zs. 


How to find the PRINCIPAL, the rate, time and interest 
being given. 


RuLE.—Divide the given interest by the interest on one 
dollar for your time and rate. 


To find the RATE, when principal, time and interest 
are given, 


RuLE.—Divide the given interest by the interest on the 
principal at one per cent. 


To find the TIME, principal, rate and interest being given, 


RuLe.—Divide the given interest by the interest on the 
principal for one day, the quotient will be the required 
time in days. 


How to Compute Time. 
RvuLE. — Subtract as in compound numbers, reckoning 
30 days to the month. 


EXAMPLE. — What is the time from January 30th, 
1869, to March 13th, 1870? 


92 
INTEREST. 


1870 3 13 

1869 1 30 

1 1 13 

Year. Month. Days. 
SUGGESTION. — When you are obliged to borrow 
from the next higher number, in subtraction of com- 
pound numbers, subtract the number in the subtra- 
hend from the borrowed number first, then add the 
number to the minuend, thus, as in above, 
30 from 30 = 0 + 13 = 13. 


Partial Payments. 


The manner of computing interest where partial 
payments have been made, has given rise to much 
litigation. The law in the different states on the 
subject very often does not clearly indicate the prin- 
ciple applicable in all cases. The aim of the law, of 
course, is to avoid usury and compound interest. 
The difficulty is in deciding whether the payment 
shall be applied to liquidate the interest or the prin- 
cipal. The U.S. rule involves compound interest, - 
as often as a payment is made greater than the interest 
then due. When more than a year intervenes, the 
U. S. rule is more favorable. 

The Vermont rule is more favorable, for there is 
no compound interest. All payments draw interest. 

We give an illustration of a problem under these 
rules, and the pupil can see the difference readily. 


The United States Rule. 


I.—The rule for casting interest when partial pay- 
ments have been made, is to apply the payment, in 
the first place, to the discharge of the interest 
then due. 


93 


INTEREST. 


II.—If the payment exceeds the interest, the sur- 
plus goes towards discharging the principal, and the 
subsequent interest is to be computed on the balance 
of the principal remaining due. 


I{I.—If the payment be less than the interest, the 
surplus of the interest must not be taken to augment 
the principal; but the interest continues on th? 
former principal until the period when the payments 
exceed the interest due, and hen the surplus is to be 
applied towards discharging the principal, and the 
interest is to be computed on the balance, as afore- 
said.—The above is the decision of Chancellor Kent ; 
Johnson’s Chancery Reports, Vol. 1, Page 17, and is 
adopted by the Supreme Court of the United States. 


EXAMPLE 1. PHILADELPHIA, May 1, 1842. 


For value received, I promise to pay to the order 
of J. THORNTON the sum of Three Hundred Dollars, 
with interest. THos. CLARK. 


The following endorsements were made on this 
note: 


1842, Oct. 16, - - - $ 60.00 
1843, March 4, - - - 17.50 
LSier Auer sitncee 26:40 
1844, Aprill, - - - 182.25. 

What was the balance due Sept. 19, 1844? 
PRRTOTITIT EGET EAI OLC). sieve ards ave rds diacele 910" 5 tes $300.00 
Mrterest to. Oct. 16; 184250. ice) wee see ewes 8.25 
PYRE ITIOU Cite tiers <9 sie ele 4 Oh Sac el ateis ok $308.25 
First Payment. ...sseecevcscccsccccesves 60.00 
New Principal.........+e0.. Mish «sts es Reto eo 


Interest from Oct. 16, 1842 to Mch. 21,1843 641 
Second Amount Padwwisic melaisie ale aalscbve sine se B24: 00 


94 


INTEREST. 


Second Amount F Sanaa tal hia oie wt arateinl elas lete eterete as el eine 


Second Payment....+..... Rh ein Gelso ie ols c ie En aa 
New Principal.....- iat bastath fot Mists. efetieice dole os PAOIeLO 
Interest from Mch. 21, 1843, to Aug. 27,1843 6.17 
Third SA morntse oniele ttoratersts cree shia is eel emiewie Pare 
Third Payment ..-.sescecessecsccovcccee 28.40 
New Principal. ..ccecsccccccccecscss 2066 $214.93 
Interest from Aug. 27, 1843, to Apr. 1, 1844 7.66 
New Principal .... «sseccecceees cose cee $222.09 
Fourth Payment ...eeseesseoees otelate.e ste ie es mia 
New Principal....+..se- Sess such dbl se Sep moUseEs 


Interest from Apr. 1, 1844, to Sept. 19, 1844 2.52 
Balance due Sept. 19, 1844. ...+ceeee.ceesf 92.86 


The following is called 


The Vermont Rule, 


And is generally applied when the time is less than 
a year. 

I.—Compute the interest on the whole debt from 
the time it was due until it is paid. . 

II.—Compute the interest on all payments, from 
the time of payment until the time of settlement. 

III.—Subtract the amount of all the payments, 
interest included, from the amount of debt, interest 
included ; the balance will be the amount due. 
KiXAMPLE 2. 

One year after date I promise to pay, to the oraer 
of D. B. JonEs, the sum of Three Hundred and Sixty 
Dollars, for value received wth use. 

SHELBY, O., Jan. 1, 1869. JOHN MILEs. 


N. B.— When no rate is mentioned, 6 per cent. is un- 
derstood. 


WEIGHTS AND MEASURES. 


AVOIRDUPOIS WEIGHT. 


PUPeS Urs) pet JEGUAl OUNCE) sesh) as) (OZ. 
TOIOUNC eS mee tee hl LLDOUD Al aniaanus ve LB. 
SI OOUIIS Etre ae 4 ce le Quarter. st aw aniat CT. 
4 quarters ... “ 1 hundred weight, cwt. 
CUMUNULeC Were a sw eet tet at see Tae sii es 


The term Avoirdupois is derived from the French avoir du poids, 
signifyiug ‘to have weight.’”” The pound consists of 7000 Troy grains. 
This weight is used for weighing almost all articles except gold, silver, 
platina, and precious stones, which are weighed by Troy Weight. 


LONG TON WEIGHT. 


28 Ibs. . . A . : “ - 5 1 quarter. 
4 quarters, or 112 lbs. , ; : : 1 hundred weight. 
20 cwt., or 2240 lbs. . : : s 1 ton. 


This measurement is nearly obsolete. It is allowed at the Custom 
House in estimating duties, and in the wholesale Coal and Iron trade. 


MISCELLANEOUS WEIGHTS. 


CEA YAS ie Sem yar teen W ASANTE TA9 Katey (8) 01} 
LOU uGaY ane nic viel os Celt in oat eS Cental: 
PEs Ne ASIN so) G.\s, 1 erp isap aul ate eT SAS 
De esr yiLiSib; ie te he ease LCouintal, 
TRE IALIS = st "ho fats tats Nett lent CCD 
PGs H OULS el cite einer” alien onie tak Darrel, 
MEM MGMULAL OTK |< coi 6 viilentel etiuke tated be Darrel. 
AMR EMEM ISTIC oihig canipritetip rratnaye via hc: GaSke 
PHOS Ali ate te hslvetecaie tee. barrel. 


96 


WEIGHTS AND MEASURES. 


THE STONE WEIGHT 


So often spoken of in English measures, is 14 lbs. 
when weighiug wool, feathers, hay, etc. ; but a stone 
of beef, fish, butter, cheese, etc., is only 8 pounds. 


HAY. 
In England, a truss, when new, is 60 lbs., or 56 


Ibs. of old hay. A truss of straw, 40 lbs. A load of 
hay is 36 trusses. 


In this country, a load is just what it may happen 
to weigh; and a ton of hay is either 2,000 lbs. or 
2,240 lbs., according to the custom of the locality. 
A bale of hay is generally considered about 300 lbs., 
but there is no regularity in the weight. 


There is no accurate mode of measuring hay but 
by weighing it. This, on account of its bulk and 
character, is very difficult, unless it is baled or other- 
wise compacted. This difficulty has led farmers to. 
estimate the weight by the bulk or cubic contents, a 
mode which, from the nature of the commodity, is. 
only approximately correct. Some kinds of hay are 
light, while others are heavy, their equal bulks vary- 
ing in weight. But for all ordinary farming purposes 
of estimating the amount of hay in meadows, mows, 
and stacks, the following rules will be found sufficient: 


As nearly as can be ascertained, 25 cubic yards of 
average meadow hay, in windrows, make a ton. 

When well settled in mows or stacks, 15 or 18 cubic 
yards make a ton. 

When taken out of mows or old stacks, and loaded 
on wagons, 20 or 25 cubic yards make a ton. 

Twenty or twenty-five cubic yards of clover, when 
dry, make a ton. 


97 


WEIGHTS AND MEASURES. 


TO FIND THE NUMBER OF TONS OF MEADOW HAY 
RAKED INTO WINDROWS. 

RuLeE. — Multiply the length of the windrow in yards by 
the width in yards, and that product by the height in yards, 
and divide by 25; the quotient will be the number of tons 
in the windrow. 

EXAMPLE. — How many tons of hay in a windrow 
40 yards long by 2 wide and 2 high? 


SoLuTion.—40 X 2 X 2= 160 —- 25=— 62. Ans. 


TO FIND THE NUMBER OF TONS OF HAY IN A MOW. 


RuLE. — Multiply the length in yards by the height in 
yards, and that by the width in yards, and divide the pro- 
duct by 15; the quotient will be the number of tons. 

EXAMPLE.—How many tons of well-settled hay in 
a mow 10 yards long by 6 wide and 8 high? 


SOLUTION.—10 X 6 X 8 = 480 —— 15 = 32 tons. 


TO FIND THE NUMBER OF TONS OF HAY IN OLD STACKS. 


Rute.—Find the area of the base in square yards, in the 
table of areas of circles; then multiply the area of the base 
by half the altitude of the stack in yards, and divide the 
product by 15; the quotient will be the number of tons. 

EXAMPLE. — How many tons of hay in a circular 
stack, whose diameter at the base is 8 yards, and 
height 9 yards. | 

SoLUTION.—50.265, area of base in sq. yards, X 44, 
half the altitude, = 226.192 —- 15 = 15.079 tons, 


TO FIND THE NUMBER OF TONS IN LONG 
SQUARE STACKS. 
Rute. — Multiply the length in yards by the width in 


yards, and that by half the altitude in yards, and divide the 
product by 15; the quotient will be the number of tons. 


98 


WEIGHTS AND MEASURES. 


EXAMPLE. — How many tons of hay in a square 
stack 10 yards long, 5 wide, and 9 high? 


SoLuTION.—10 X 5 X 44 = 225 —- 15 = 15 tons. 


TO FIND THE NUMBER OF TONS OF HAY WHEN TAKEN 
OUT OF MOWS OR OLD STACKS. 

RULE. — Multiply the length of the load in yards by the 
width in yards, and that by the height in yards, and divide 
the product by 20; the quotient will be the number of tons. 

}iXAMPLE.—How many tons of hay can be taken 
from an old stack, in a load 6 yards long by 3 wide 
and 3 high? 

SOLUTION.—6 X 3X 3= 64 — 20 = 2,5 tons. 

These estimates are for medium sized mows or 
stacks. Ifthe hay is piled to a great height, as it 
often is where horse hay-forks are used, the mow will 
be much heavier per cubic yard. 


AN EASY MODE OF ASCERTAINING THE VALUE OF A 
GIVEN NUMBER OF LBS. OF HAY, AT A GIVEN PRICE 
PER TON OF 2000 LBS. 

RuLe.—Multiply the number of pounds of hay (coal, or 
anything else which is bought and sold by the ton) by one 
half the price per ton, pointing off three figures from the 
right hand; the remaining figures will be the price of the 
hay (or any article by the ton). 

ExaMPLE.— What will be the cost of 658 lbs. of 
hay, at $7.50 per ton? 


SOLUTION. — $7.50 divided by 2 equals $3.75, by 
which multiply the number of pounds, thus: 
658 
$3.75 
$2.46||750. Ans. 
Nors.—38.75 is §; therefore } of 658 = 82, and # is 3 times 
82 or $2.46. 


99 
WEIGHTS AND MEASURES. 


A BALE OF COTTON 


In Egypt is 90 lbs.; in America a commercial bale 
is 400 lbs. ; though put up to vary from 280 to 720, 
in different localities. 

A bale or bag of Sea Island cotton is 300 lbs. 


WOOL, 


In England wool is sold by the sack or boll, of 
22 stones, which, at 14 lbs. the stone, is 308 lbs. 

A pack of wool is 17 stones and 2 lbs., which is rated 
as a pack load fora horse. It is 240 lbs. <A tod of 
wool is 2 stones of 14 lbs. A wey of wool is 64 tods. 
Two weys, a sack. A clove of wool is half a stone. 


A quarter of corn or other grain sold by the bushel 
is eight imperial bushels, or quarter of a ton. 


A ton of liquid measure is 252 gallons. 


BUTTER 


Is sold by avoirdupois weight, which compares with 
troy weight as 144 to 175 ; the troy pound being that 
much lighter. But 175 troy ounces equal 192 of 
avoirdupois. 

A firkin of butter is 56 lbs.; a tub of butter is 
84 lbs. 


THE KILOGRAMME OF FRANCE 


Is 1000 grammes, and equal to 2 lbs. 2 ozs. 4 grs. 
avoirdupois. 


100 


WEIGHTS AND MEASURES. 


COAL. 


A cubic foot of anthracite coal, before it is pre- 
pared for domestic use, will on an average weigh 
about 93 Ibs. When broken for the market it will 
average about 04 lbs. 


TO ESTIMATE THE WEIGHT OF COAL IN ANY 
GIVEN SPACE. 

RuLE. — Multiply the contents in cubic feet by 54, for 
anthracite, or by 50, for bituminous coal, and the product 
will be the weight in pounds. 

1. How many tons of anthracite coal, of 2240 Ibs. 
each, can be stored in a bin 28 ft. long, 20 ft. wide, 
and 4 ft. deep? 

ANALYSIS. — 28 X 20 X 4 X 54 —>- 2240 = 54 tons. 

2. How many pounds of bituminous coal in a car 
30 ft. long and 7 ft. wide, the depth of the coal being 
16 in.? 

3 How many pounds of anthracite coal can be 
placed in a cart which measures 6 ft. in length, 44 ft. 
in width, and 16 in. in depth? 

4, I wish to build a bin in my cellar to hold 8 tons 
of anthracite coal, 2240 lbs. to the ton ; I have made 
the length 12 ft., and the width 10 ft. ; what must be 
the height of the bin? 

5. How many pounds of bituminous coal can be 
stored in a space 50 X 50 X 123 ft.? 

6. How many tons of anthracite coal, 2000 Ibs. to 
the ton, can be stored in a yard which measures 
60 ft. in length, and 380 ft. in width, the depth of the 
coal being 6 ft.? 

7. A dealer purchases 1500 tons of anthracite coal, 
2240 lbs. to the ton, which he wishes to store in an 
inclosure 100 ft. long, and 80 ft. wide ; what will be 
the depth of the coal? 


101 


WEIGHTS AND MEASURES. 


TABLE OF AVOIRDUPOIS POUNDS IN A BUSHEL. 


The following Table shows the weight of a bushel as prescribed by statute, in the several States named. 
2 rae abe Si a 
wees 3 nN E wT AN| . 8 aa ES 
S/S ,| 3 <} ei Si. / 81S] 2] % 84 Seles 
COMMODITIES. F/R] |g 21S) STE Seis] P88 ei S iS] Ry ais 
SPSS TSE S| SPSL RT STS] ST si Sia gs] Si 818/81 815 
PPE PEIN SB] SLRS Sis lglagiciaterecie stsisis ies 
SIS/ISIE/ SIS ISISIS/S ISIS [SIS FSIS IS {S]alRIN ARIA 
Barley, 5 : . |50|.. 148/48] 48]48 | 32]... | 46] 48] 48) 48] .. | 48] 48 | 48 1°48 | 46 | 47 | .- | 46 | 48 | 45 
Beans, é 5 | eres terse GOI OOH GO aHO0 © | sere dee) Sore wesw] cote AOUL|: ore lee pmeti-etote| O-culae eit) cast lene | Mora ese emcee eete 
Bituminous Coal, Seen eee Zu O10 -livconll soo Borsetebetee’| eee |" GU|Se1eb [rors] dead |-eedl vers], erecta Oll| scart arom oteial ls 
Blue Grass Seed, See | ee ree Sd ah | DA als ae lees | a | Sater al Valeo celle col lster4eten| nee || tere. |eele dawre nines tlaete ti te’s 
Buckwheat, : . 1401451 40| 50/52) 52 | .. | .. | 46 | 42] 42152) .. | 50150) 43] .. | 42] 48] .. | 46 | 42 | 42 
Castor Beans, . Reis cr eee 46a AO nhs eine [eich weet sere "| tern ater |) 4 Ons letemalieeren | ares Peres | ere [tie waa erei| evens eetemlne on fret 
Clover Seed, . . | ee |... |60|60]60)60 | ..]..|.- | 60] 60]60]..| .. | 64] 60] 60] 60|..|.- | -- | 60] 60 
Dried Apples, . ae [etfs PAO Ge) OA | Aneel ea) — eels ck | CSc aon aed decree ew [cores Lpoean cies [52Onl Foren eee leewn| ao: |e 
Dried Peaches, . Fe eo er AB Se 1 Bat L we altos he ochre 1S SB°SS Pe ee seth oa Le | Teesk. wet Piva eed ame 
Flax Seed, r hice dose 1 DO ROGA DO | G6 cl wel Sel ee les ed-ee | OG ies lan 1 OO 100-1 BG Lise ced en |-een Dodie. 
Hemp Seed, ° eat es eed 4 ta | Ae oleae te Lae nae [eae oleee ne at | veer) ouellanven [ae em] ween] hematite tart kere 
Indian Corn, O . |52/56| 52) 56156156 | 56].. | 56 | 56] 56] 52] .. | 54 | 56 | 58] 56 | 56 | 56 | .. | 56 | 56 | 56 
Indian Corn in ear, . ee ee 70 68 68 cael oe ee oo ee ee ee ara oe pee ee ee ee ee ee ee ee ee 
Indian Corn Meal, eae licey eal eee | ASI CVO lesrect fe. | uroyein)|| Os sO retour lucien [ate om | eran leer | athe Sen teever| Semel OO Wee oe eater |aere 
Cates . | 32 | 28] 32] 32] 35 | 334] 32 | 30 | 30 | 32 | 32 | 35] 30| .. | 30 | 32] 32] 34] 32] .. | 32/32) 36 | 
Onions, : 7 ET BTL Bil BT ABT = loon, hea OS Pact sa OF aettes lise bees Iueebicasei | OOM sr eaieeeeboumas 
Potatoes, . - |e. |60/60| 60/60/60 |..|/60|-.]|..]..|60] 60] .. | 60/60] .. | 60] .. | 60|60) 60/60 | 
Rye, . . . . 1541561541561 56156 | 82].. | 56) 56156) 56]..|.. | 56] 56 | 56 | 56 | 56] .. | 56 | 56 | 56 | 
Rye Meal, . y eee eS dee | eet Sea) LOO eed feet | tae ikea |) seat Se arke aoe heen bleh ae ere re 
Salt, . ; Se relies seen DOs DOA -DO. Worst gteclite ss |e ietl| Meret OO Mlnertal mates pict MOO ll worte leerattecemnrese tae ee Jee 
Timothy Seed, . Bee cee (Dea a cS | AB alae arte eull eee Meee |e ce AD dice cele ory mote eed lerorom aneteke  Pevcaeal gerald On ase 
Wheat, oe i . |60]5C 160] 60] 60}60 | 60] .. | €0| 60] 60 | 60] .. | 60 | 60 | 60 | GO| GO | 6D] .. | 60 | 60 | 60 
Wheat Bran, . pre hee atria’ | Oat ay ov tO OO | tas te ee ares’ £20 bee soliwinc Gacy = AMOR aeauey cok. OUTS. ea eee 


102 


RATLROAD FREIGHT, 


GROSS WEIGHTS. 


The articles named are billed at actual weights, if possible, but 
usually at the weights in the table below when it is not convenient to 


weigh them. 


Ale and Beer ......320 lbs. per bbl. | High wines.........3501bs. per bbl, 
Kf Pre a ae 170 ** } * | Hung’n Grass Seed 45 ‘“ bu. 
ss Oe acre eta LOO ie Aare Wy Shoat GLO caretaite wi tptetey 2 200 hee bp 

Apples, dried...... 24 ‘ bu. | Malt..... Gta dare S806) 2 ba. 

66 DVOON | Asia's 6 SOO Jute ‘¢ | Millet..... eosseencs a res ‘f 
“ Shee taele be 150, “bbl. | Nails ..... Bs p.ule ein «108.7 FF Kee 
Barley......« se oy EO oe SU GRITS) | CORE Wart orale diareals ihe. 6 Aare FE Pee 0s 
Beans, white....... 60) 5,556 56 PUN CULES wie cieiatata Nal Jicisiatatata 400 * bbl 
a castor .....- 46 ost FF Onions. 3. con's aie Gattis Lenn bu. 

BOGE 6 66h obese veee.520.° **) ebb Peaches, dried...<c.a0s.)- BF 

BEAN e venue con's ea ae 20 nie DUP Perk s.cpedencserees 320 ito DD Ly 

Brooms..... soceeee 40 doz. | Potatoes, common.150 ad 

Buckwheat ........ 52 § bu. “6 6 60: ees bu. 

Cider s... <ssseeec.d00 |)" Db. as sweet.... 55 § 

Charcoal..... ic 277s DUH aRV Oa adistiegie mein ox atae DOWN. Be 

Clover Seed........ 60 “ Fl Salt, fine... depress. BG canes ce 

Corns aucisecrs aeoesie Dog ies: G CO telseisiaees Dee sOUO Mi Cama D ple 

$e Sn CAP saben ciel dy (€) ) COATSO 204.0450. SD0) 2% 6 
SS Te IVECBL iois\ncc ole ciain eer We We ' in gacks.....-200 sack. 
“6 La aera eeseaad. ) & bbl. | Eimothy Seed)..... 46 16) “ibn. 

Eggs... wcceerseee ZOny se <6 | Turnips.......s.0- a Sale 

Fish ....sesee. ovewed0O. 7 68 “| Vinegar...... veeeeedd0 «6 ~~ ODDL. 

Plax Seed.........: BG mass DUW Ie WIDCRE stare lelsvcre rete teil GOpas bu 

Milour. ) A: 200 bbl. | Whiskey ...... «ene BOO (Ml BbL 

Hemp Seed......-.. 44.“ bu. | One ¢ox weight is 2000 lbs. 


Nore. — From 18,000 to 20,000 lbs. is considered a car-load in most 
places, each car itself also weighing about 20,000 lbs. 


To Estimate Grain Crops per Acre. 


Frame together four light sticks, measuring exactly a foot square in- 
side, and with this in one hand, walk into the field and select a spot of 
fair average yield, and lower the frame sguare over as many heads as 
it will inclose, and shell out the heads thus inclosed carefully, and 
weigh the grain. Itis fair to presume that the proportion will be the 


43,56Cth part of an acre’s produce. 


To prove it go through the field 


and make ten or twenty similar calculations, and estimate by the mean 
of the whole number of results. 
make a closer calculation of what a field will produce than he can by 


guessing. 


It will certainly enable a farmer to 


108 


The Metric System of Weights and 


Measures, 
With their Equivalents According to the System in Use. 


MEASURES OF LENGTH. 


Metric Denominations and Values. Equivalents in Denominations in use. 
Myriameter..... = 10,000 meters...... = 6.2137 miles. 
Kilometer ......= 1,000 meters...... — 0.62137 mile or 3,280 ft. 10 in. 
Hectometer..... 100 meters..... - — 828 feet and 1 inch. 
Dekameter...... 10 meters...... = 393.7 inches. 
Meter... .j..s eee 1 meter....... = 39.37 inches. 
Decimeter...... -1 of a meter .... = 3.937 inches. 
Centimeter ..... -01 of a meter .... = 0.3937 inch. 


No Wt) UU 


Millimeter...... = .001 of a meter .... = 0.0394 inch. 
MEASURES OF SURFACE. 
Hectare ........ = 10,000 square meters......... = 2,471 acres. 
AlOvcewcaccsc sae = 100 square meters......-..—= 119.6 square yards. 
Centare......... = 1 square meter.......+-. = 1,550 square inches, 
MEASURES OF CAPACITY. 

Names. No. Liters. Cubic Measure. Dry Measure. Wine Measure 
Kiloliter.. =1,000.... = 1 meter... = 1,308 cub. yds.. = 264.17 gails. 
Hectoliter.= 100....—= .1 meter... = 2 bu. 3.35 pks.. = 26.417 galls. 
Decaliter. = 10....—=10decim... = 9.08 quarts.....—= 2.6417 galls. 
Liter .....= 1....= 1 decim... = 0.908 quart. .... = 1.0567 qts. 
Deciliter..= .1....= .1 decim... = 6.1022 cub. in.. = 0.845 gill. 


Centiliter. .01 ... = 10 centim.. — 0.6102 cub. in.. — 0.338 fi’d oz. 
Milliliter. = .001....= 1centim.. = 0.061 cub. in... = 0.27 fluid dr. 


WEIGHTS. 
Weight of what quan- 
tity of water atmaxi- Avoirdupois 
Names. No. Grams. mum density. Weight. 

Millier or tonneau.. = 1,000,000.... = 1 cubic meter.... = 2204.6 lbs. 
Ci talestateesles sia 's 100,000.... = 1 hectoliter ...... = 220.46 lbs. 
Myriagram......... 10,000...- = 10 liters ........... = 22.046 lbs. 
Kilogram or Kilo .. 1,000.... = 1 liter ......0..++- == 2.2046 Ibs. 


100.... = 1 deciliter ........= 3.5274 oz. 
10.... = 10 c. centimet...... = 0.3527 oz. 
= lc. centimet..... - — 15.432 grs. 
1....= 1c. centimet...... = 1.5432 gra. 
01.... = 10c. millimet....-. = 0.1543 gr. 
001....—= 1c. millimet... .. = 0.0154 gr, 


. 


a el 
i 


Hectogram.....+.. 
Dekagram ....seee- 
GYAM .ececccecceess 
Decigram....... «- 
CentigraM eeeeecere 
Milligram cccceeeess 


104 
WEIGHTS AND MEASURES. 


LIQUID or WINE MEASURE. 
Avpills (oi} io Go  POCMALGl a pint, Silt i. toe tape 
2 pints . EAS LUA ho ot Te 
A quarts, 00 Cee eed OL Oh) sire male 


The standard liquid gallon contains 231 cubic inches. — In the old 

tables were given 31% gallons = 1 barrel; 63 gallons = 1 hogshead ; 

hogsheads = 1 pipe; 2 pipes =1tun. These are now obsolete, because 
scarcely any cask holds exactly the same amount of liquid. 


Square Cisterns. 


RuLE.—Multiply the length in inches by the width in 
inches, and that product by the depth in inches, and divide 
the whole product by 281. 

EXAMPLE.—How many gallons in a cistern 6 feet 
long, 3 feet wide, and 4 feet deep. 

SOLUTION.—72 inches in length X 36 inches’ width 
X 48 inches’ depth = 124416 —— 231 = 538.59 gals. 


Approximate Method for Square Cisterns. 


RULE. - Divide the solid contents in feet by 4, and you 
will have the contents in barrels of 314 gallons each. 

Take the preceding example: 6X3X4—-4—= 
18 barrels. 

Notse.—This rule gives only the approximate contents, 
and is based on the following: There are 314 gallons ina 
bbl., and 231 cubic inches in a gallon, which, divided by 
1728, the number of cubic inches in a foot, we have about 
4, and for ordinary calculation often gives the correct 
quantity for many practical purposes. 


Approximate Method for Round 
Cisterns. 

RuLE. — Multiply the depth in feet by the diameter in 
feet, and that product by 14, and you have the number of 
barrels of 314 gallons each. 

ExAMPLE.—How many barrels in around cistern § 
feet deep by 5 feet in diameter? 

SoLution.—8 X 5 X 14 = 60 barrels. 


105 


WEIGHTS AND MEASURES. 


Capacity of Round Cisterns or Tanks. 


Tabular view of the number of gallons contained in the 
clear between the brickwork for each ten inches in depth: 


DIAMETER. GALLONS.|DIAMETER. GALLONS. 
2 feet equal...ee.sececeees 19 8 feet équal........./.55. 313 
DT et ceegaswceces 30 SAFO Ee radleswstivag tees 353 
ee EET aveuocvegeuscues 44 Dee EIS EE asec waeneuss 396 

Be) EE Sen ecceucs sees 60 DAs SRS ns dmae be ate scale SOR 
ih ie da cpekesad esd 78 LO eee BOTs Siew eae Maly i'd 489 
BO, EE ie ca vesea diene 97 AN oN eye EERE NS Pap ee 592 
Bae ee eebevess somuenbwe 122 Lt od webeaae adea's 705 

BA? NO as ce cepen sees ds 148 DBS Eee news 5 we ong 0 827 
Ge oe secdecedvcuees 176 Late th Sadatsie eepe 959 
Oho al) ie ean ceaseessccass 207 Fy WN Ea vec ow comers oeaie's 1101 
Deel PE Scasneds sven ness 240 2 oS i TMG idee Werd aie de 1958 
78 LS RP ar err ee 275 257, Oe OM see e esr ae’s cess 3059 


Circular Cisterns. 


RuLe.— Find the area of the circle by our table of multi- 
ples; then multiply this by the depth in inches, and divide 
this product by 231. 


EXAMPLE.—A cistern is 8 feet in diameter by 5 feet 
deep. How many gallons does it contain. 

SOLUTION. — Multiply the area by 96 inches, the 
diameter = 7238.2 X 60 inches, the depth = 43429,20 
—- 231 = 1880 gallons. Ans. 


Another Rule for the Measurement of 
Cylindrical Cisterns. 


Take the length, width and depth in feet; multiply these 
together, and the product by 1865; cut off four figures on 
the right, and the result will be the contents in barrels. 


EXAMPLE. — Find the contents of a cistern 6 feet in dia- 
meter and 9 feet deep. Six feet, the length, multiplied by 
6 feet, the breadth, and the product by 9, the depth, gives 
324, which multiplied by 1865, and four figures cut off, gives 
60 barrels and a decimal. 

In this case we consider the diameter as being both 
length and breadth. The reason of the rule is this: acyl- 
inder one foot in diameter and one foot long would measure 
1865 ten-thousandths of a barrel. A cylinder 9 times as 
long would contain 9 times as much, and 6 times as wide, 
6 times as much as that. The number 1865 is easy to re- 
member, as it corresponds with the number of a year. 


106 


WEIGHTS AND MEASURES. 


ALE or BEER MEASURE. 


2 PINtS (Dts neers | OUARUUMCIIAL ER tote tsar ee 

Ai Quarts i's. hits Leni ee MOURA NON UN wh scent Dake 
S6vallons.) ph Gusiee Ue RnebarTe Let ec ar Oks 
D4 gallons) ssi; 8b 4) Mio ea ROeShead, wit a phd, 


Ale or Beer Measure is used in measuring ale, beer, etc. The gallon 
consists of 282 cubic inches. 


TROY WEIGHT. 


24 grains (gr.) . . equal 1 pennyweight, dwt. 
20 -pennyweights -(./ 00 Wie OU Ceal \y ui lecunaiie 
LZ OUNCES li i shee ateEe ODOT Cy Ue ini eal ae 


The term Troy is said to be derived from 7yoyes, the name of a town 
in France, where the weight was first used in Europe. The symbol 
(oz.) is from the Spanish word ovzza, for ounce, and (lb.) from /ibra, 
a pound. 


APOTHECARIES’ WEIGHT. 


20 grains (gr.) . . equal 1 scruple, 
GUSCTUDIES loans ts Marte e mur CLod arte 
Sidramsy (ov don ge A CORN COLts 

LA tORNCES Mish ete es OLN « 


This weight is used in mixing and retailing medicines. The pound 
is the same as the pound Troy. 


= ON a Ww 


TIME. 


60 seconds (sec.) . . equal 1 minute, . . m. 
60 minutes 7 EL Aer ae ae Onn etre airy 
PAN GUESE . esige tn bear ta coe Maal arora a tv CaaS 
PVC AYS Lich ants Ug Laue Oe NP aie ames tr as Mt bee 
LI WRER SH re UI a 2) ec SOARS DOM at ter) TO 
SRAWECKS 401) he He N tant” ae ROARS trix Talc nig 
12 calendar months iyo yh veatn ye. 6 eae 


107 


SIGNS AND ABBREVIATIONS USED BY PHYSI- 
CIANS IN PRESCRIPTIONS, MEDICAL 


BOOKS AND 


R. Receipt. Take. 


A., ia, ana, utriusque. Of each. 
Ib Libra. A pound. 

z Uncia. An ounce. 

i 3 Fiutduncia. .A fluid ounce. 

5 Drachma. A drachm. 

14 Fluidrachma. A fluid drachm. 
D  Scrupulum. A seruple. 

ss. Semissis. Half. 

iss. Sesgui. One and a half. 
Abdom. Abdomen. 

Abs. Febr. <Adbsente febre. In 


the absence of fever. 

Ad., or Add. Adde, or Addatur. 
Add, or let there be added. 

Ad Lib. Ad libitum. At pleasure. 

Altern. Hor. A lternis horis. Every 
other hour. 

Aq. Aqua. Water. 

Aq. Bull. Aqua dulliens. 
water. 


Boiling 
communis. 


Hot 


wh Comm. Aqua 

Jommon water. 

Aq. Ferv. Agua fervens. 
water. 

Aq. Font. 
water. 

. B.A. Balneum arene. 
bath. 

Bib. Bibe. Drink. 

Bis Ind. BSzs zzdies. Twice daily. 

Bol. Bolus. A large pill. 

Bull. Budiiat. Let it boil. 

B.V. Balneumvaporis. A vapor 
bath. 

Cap. Cafiat. Let him take. 

Chart. Chartula. A small paper. 

Cochl. Cochleare. A spoonful. 

Col. Cola. Strain. 

Collyr. Collyrium. Aneye water. 

Comp. Compositus. Compound, 

C.,or Cong. Congius. A gallon. 

Coq, Cogue. Boil. 

Cort. Cortex. Bark. 

C. M. Cras Mane. 
morning. 
- N. Cras nocte. 
night. 

Crast. Crastimus. For to-morrow. 

D. Detur. Let it be given. 

Decub. Decudbitus. Lying down. 

De D. in D. De die in diem. From 
day to day. 

Dig. Digeratur. Let it be digest- 
ed. 


Dil. Dzlutus. Diiute. 
Dim. Dimidius. One-half. 
Div. Divide. Divide. 
Enem. “Anema. <A clyster. 


Spring 
A Sand 


Agua fontis. 


To-morrow 


To-morrow 


JOURNALS. 


F. Fiat. Letit be made. 
F. Pil. Fiat pilula. Make into 
a pill. 


Feb. Dur. Febre durante. During 
the fever. 

Fl. Fluidus.' Fluid. 

Gr. Granum. A grain. 

Gt. Gutta. A drop.—Gtt. 
te. Drops. 

Guttat. Guttatim. By drops. 

Hor, Decub. Hora decubitus. 


bed-time. 
H.S. Hora somnt. At bed-time. 
A pound 


Gut- 
At 


Libra. 


Liquor. 

Misce. Mix. 
Man. Minipulus. 
Mic. Pan. 


A handful. 
Mica Panis. Crumb 
of bread. 


Min. Minimum. The sixtieth 
part of a drachm by measure. 
Mist. Mistura. A mixture. 
Muce Mucilago. Mucilage. 
O. Octavius. <A pint. 
Ol. Oleum. Oil. 
Omn. Hor. Ovni hora. 
hour. 
Omu. Man. 
morning. c 
Omn nocte. Every night. 
Oz. Uncia. An ounce. 
P. i. Partes equales. 
parts. 

Pil. Pilula. A pill. 
P.R.N. Pro ve nata. 
sion may require, 

Pulv. Pxlvis. A powder. 

Q.S. Quantum sufficit. As much 
as is sufficient. 

Rad. Radix. Root. 

Rep. Regetatur. Let it be re- 
peated. 

8S. Signa. Write. 

8S. A. Secundum artem. Accord- 
ing to art. 

Seed. 


Sem. Seven. 
Sz non valeat. If it 


Si Non Val. 
If there be 


Every 


Omni Mane. Every 


Equal 


As occa- 


does not answer. 
Si Op. Sit. Sz opus sit. 
need. 
Sing. Singulorum. Of each. 
Solv. Solve. Dissolve. 
Sp. Spzritus. Spirit, 
Sum. Szsmat. et him take. 
Sp. Vin. Spiritus vini. Spirit of 
s ee 
, 3h YUPUS. 
Ty. Tinct. 4 
Vs. Veneesectio. 


Syrup. 
Tinctura. Tincture. 
Venesection. 


108 


WEIGHTS AND MEASURES. 


DRY MEASURE. 


2 pints (pt-)\ <. vequaleliquatts gy. ets. cl ts 
ofa] bE Nai: Semel Cipaih) Vik yey nse) gq Ne Mi 0 (6)! 
ASDECKS isin You euitad Ye LUISE: Wate ie cnEG 
oO bushels (ah eae Chaloron ecu auch 


Dry Measure is used for measuring grain, fruit, coal, etc. 


To Measure Corn in the Crib, 


When the crib ts equilateral. 


Rutle.—Multiply the length in feet by the breadth in 
feet, and that again by the height in feet, which last prod 
uct multiply by .63 (the fractional part of a heaped bush- 
el in a cubic foot), and the result will be the heaped bush- 
els of ears. For the number of bushels of shelled corn, 
multiply by 42 (two-thirds of .63), instead of .63. 


EXAMPLE.— Required the ‘number of bushels of 
shelled corn contained in acrib of ears, 15 feet long 
by 5 feet wide, and 10 feet high? 


15, length, X 5, width, X10, height, = 750 cubic ft. 
Then 750 X .63 = 472.50 heaped bushels of ears. 
Also 750 X .42 = 3815 bushels of shelled corn. 


In measuring the height, of course the height of 
the corn is intended. And there will be found to be 
a difference in measuring:corn in this mode between 
fall and spring, because it shrinks very much in the 
winter and spring, and settles down. 


When the crib ts flared at the sides. 

Rule.—Multiply half the sum of the top and bottom 
widths in fee¢t by the perpendicular height in /ee¢, and that 
again by the length in feet, which last product multiply by 
.63 for heaped bushels of ears, and by .42 for the number 
of bushels of shelled corn. 


109 


WEIGHTS AND MEASURES. 


LONG MEASURE. 


P2einches: (ins) » or, CUAL LE foot Mie cats Les 
PCC Me Net ey cgi) DAOVATC a hrwemial) an evCle 
DEVELO Ga ee ca) elle: vg) yh’ LaTOOs Use ote ca TO. 
PUEOUS TAN Matias i? 1 sy A RULIOMS sey) « cEUlS 


PLUTONS Gal he cer za. ses dy MEL OS Ge) to es” te pad 


The yard is the standard unit of length. It is formed by dividing a 
pendulum, which vibrates seconds in a vacuum, at the level of the sea, 
at the latitude of London, into 391,393 equal parts and taking 360,000 of 
these parts. From this unit all other measures and weights are derived. 


NotTe.— Cloth measure is practically out of use. In meas- 
uring goods sold by the yard, the yard is divided into salves, 
fourths, eighths and sixteenths. At United States Custom 
Houses, in estimating duties, the yard is divided into zexths 
and hundredths. 


For Measuring Heights and Distances. 


PAC HOS EM Ne Winona, Nt eh aioe eed \ La TK. 
LAAN gil SEU PY TLR Lenya ai ees ithe DAU 
ee RR Renata tg eee Lc SUDAN. 
DRT PlCeU sau yor iale eb era my eat) by ut PACE, 


Mariner’s Measure. 


Table used by Mariners in calculating distances on 
water, and the speed of vessels. 


SRI CHESN yt WAG re CaM: Shani, 
SISpans Or G Les c ia. ype ane eaeLO 
MULL NOMS i micah crt eM en ne CAUIG IS LeCUSEn, 
Taccabless tae. eee male or. knot: 
GOSGiemealt ys ia! ay som (echt alita el gtatt ANU LeRoy 
aanuess on knots. .t (cued sou b leagne. 


Notre. — The number of knots of the log line run off in 
half a minute indicates the number of knots of distance a 
vessel goes per hour. A nautical mile or knot equals near- 
ly 1 statute miles. 


110 


WEIGHTS AND MEASURES. 


Surveyor’s Long Measure, 


For measuring boundaries of land, areas, vatlroads, canals. 


73975 inches BE cea Ol iad Fai ks 
25 links, iad Hib caret 

4 rods, . so ‘Chait, 
80.chains, |) ova, vy bame noe een etoahee om tial ng te 


10 chains long by 1 broad, or 10 sq. chains, 1 acre. 


GUNTER’S CHAIN, which is the unit of measure 
used by surveyors, is 66 feet long, consisting of 100 
links. 

Measurements are recorded in chains and hun- 
dredths. Latterly a steel measuring tape 100 feet 
long, with each foot divided into tenths, is used by 
engineers as a substitute for the cumbersome chain. 

Norte. — By scientific persons and revenue officers the 
inch is divided into fenths, hundredths, etc. Among 


mechanics the inch is divided into ezghths. The division of 
the inch into 12 parts, called lines, is net now in use. 


A standard English mile, which is the measure we use, 
is 5280 feet in length, 1760 yards, or 820 rods. A strip one 
rod wide and one mile long is two acres. By this it is © 
easy to calculate the quantity of land taken up by roads, 
and also how much is wasted by fences. 


TABLE 


For Geographical and Astronomical Calculations. 


1 Geographic mile, . . . . 1.15 statute miles. 

3 * Eos ial eM ase! (ke LN OT 
60 - “* or 69.16 statute miles, 1 degrec. 
360 Degrees, . . . Circumference of the earth, 
Nore. — The earth’s circumference is 24,8551¢ miles, 


nearly. The nautical mile is 60753 feet, or 795$ feet longer 
than the common mile. 


111 


WEIGHTS AND MEASURES. 


Shoemaker’s Measure. 


Gebaneyeorns, on sizes. os) (heute dig inehs 


Number one, children’s measure, is 48 inches, and 
every additional number calls for an increase of 
4 of an inch in length. Number one, adults’ measure, 
is 84 inches long, with a gradual increase of $ of an 
inch for additional numbers, so that, for example, 
number ten measures 114 inches. This measure cor- 
responds to the number of the /as¢, and not to the 
length of the soée. 


Surface or Square Measure, 


USED IN ASCERTAINING THE EXTENT OF SURFACES, 
SUCH AS LAND, BOARDS, PLASTERING, PAVING, ETC. 


TABLE. 
144 Square Inches (sq. in.) 1 square foot, sq. ft. 
9 Square Feet,’ . , .. 1 square yard,.sq. yd. 
304 Square Yards, . . . 1sq.rodorperch, sq. rd.; P. 
160 Square Rods, ..,... l acre, A. 
GAO CACHES 14: vi de inuisiee . . 1 square mile, sq. mi. 


Measure 209 feet on each side, and you have a 
square acre within an inch. 

Note. — The following gives the comparative size, in 
square yards, of acres in different countries : 

English acre, 4840 square yards; Scotch, 6150; Irish, 
7840; Hamburg, 11,545; Amsterdam, 9722; Dantzic, 6650; 
France (hectare), 11,960; Prussia (morgen), 3053. 

This difference should be borne in mind in reading of 
the products per acre in different countries. Our land 
measure is that of England. 

ARTIFICERS estimate their work as follows: 

By the sguare foot ; as in glazing, stone-cutting, etc. 

By the sguare yard, or by the sguare of 100 square 
feet ; as in plastering, flooring, roofing, paving, etc. 


112 


WEIGHTS AND MEASURES. 


CUBIC MEASURE. 


‘TABLE. 

1728 cubic inches (cu. in.), . . 1 cubic foot, cu. ft. 
DArCubIO TCC OL eames 1 cubic yard, cu. yd. 
40 cubic ft. of round timber ish CERAOe TOA 
50 cubic feet of hewn timber aay 
16 .cobicgdeetenn csr wa. 1 cord foot, cd. ft. 

H iS iter at Va 1 cord of wood, Cd. 
SAR CUDIG TCEL ace wee erane cnt 1 perch or stone, or 


masonry, Pch. 


TO FIND THE CUBIC CONTENTS OF ANY SOLID BODY. 

RULE. — Multiply the length by the breadth, and that 
product by the thickness. 

Notres.—1. A load of earth contains a cubic yard, 
and weighs about 3250 lbs. 

2. Railway and transportation companies estimate 
light freight by the number of cubic feet it occupies ;. 
but heavy freight is estimated by weight. 

8. A pile of wood 4 feet wide, 4 ft. high, and 8 ft. 
long, contains 1 cord; and a cord foot is 1 foot in 
length of such a pile. 

4. A perch of stone or masonry is 164 feet long, 
14 feet wide, and 1 foot high, and contains 24# 
cubic feet. 

5. A brick is usually 8 inches long, 4 inches wide, 
and 2 inches thick; hence 27 bricks make a cubic 
foot. 

6. Joiners, painters, and masons make no allow- 
ance for windows, doors, etc. In some places it is 
customary to allow for one-half the opening. Masons 
make no allowance for the corners of the walls of 
houses or of cellars. The size of a cellar is estimated 
by the measurement of the outside of the wall. 


113 


WEIGHTS AND MEASURES. 


Ton Weight and Ton Measure. 


A ton of hay, or any other coarse bulky article 
usually sold by that measure, is 20 gross hundreds, 
that is 2249 lbs. But in many places it has become 
the custom to count only 2000 lbs. for a ton. In 
freighting ships 42 cubic feet are allowed to a ton; 
in the measurement of timber 40 solid feet if round, 
and 50 if square, make a ton. 


MASONRY. 


A perch of stone is 24.74 cubic feet ; when built in 
the wall, 22 cubic feet make 1 perch, 22 cubic feet 
being allowed for the mortar and filling. 

Three pecks of lime and four bushels of sand to a 
perch of wall. 


To find the number of perches of stone in walls. 


RuLE.—Multiply the length in feet by the height in feet, 
and that by the thickness in feet, and divide the product 
by 22, and the quotient will be the number of perches of 
stone in the wall. 

EXAMPLE.--—-How many perches of stone contained 
in a wall 40 ft. long, 20 ft. high, and 18 inches thick ? 

SoLuTion.—40 ft., length, X 20 ft., height, X 14 ft., 
thickness, = 1200 —- 22 = 54.54 perches. Ans. 


Note.—To find the number of perches of masonry, divide 
the product, as above, by 24.75, instead of 22. 


Brickwork. 


The dimensions of common bricks are from 7? to 
8 inches long, by 44 wide, and 2} thick. Front bricks 
are &4 inches long, by 44 wide, and 24 thick. 

The usual size of fire bricks is 94 inches long, by 
43 wide, by 23 thick. 


114 


WEIGHTS AND MEASURES. 


Twenty common bricks to a cubic foot when laid; 
15 common bricks to a foot of 8-inch wall when laid. 


To find the number cf common bricks in a wall. 


RuLE.— Multiply the length of the wall in feet by the 
height in feet, and that by its thickness in feet, and that 
again by 20, and the product will be the number of bricks 
in the wall. 

EXAMPLE. — How many common bricks in a wall 
40 feet long by 20 feet high and 12 inches thick? 


SoLuTIon.—40 ft., length, X 20 ft., height, X 1 ft., 
thick, X 20 = 16,000. Ans. 


Note.—For walls 8 inches thick, multiply the length in 
feet by the height in feet, and that by 15, and the product 
will be the number of bricks in the wall. 

When the wall is perforated by doors and windows, 
or other openings, find the sum of their cubic feet by 
severally multiplying their lengths and widths and 
thicknesses in feet together, and deducting the whole 
from the cubic contents of the wall, including the 
openings, before multiplying by 15 or 20, as above. 


Laths. 


Laths are 14 to 14 inches wide by 4 feet long, are 
usually set 4 inch apart, and a bundle contains 100. 


Short Approximate Method for Estimating 
Stone Work. 


Rute. — Multiply the length in feet by the height in feet 
by the thickness in feet, and that product by 4, cutting off 
the two right-hand figures. 


EXAMPLE.—How many perches of stone in a wall 
40 ft. long, 20 ft, high, and 2 ft. thick? 


40 X 20 X 2 X 4 = 64.00, or 64 perches. 


CARPENTERS’ ESTIMATES. 


SHINGLES are usually 16 inches long and, on an 
average, very nearly 4 inches wide. 


They are put up in bundles 20 inches wide and of 
24 courses. Four such bundles contain 1000 of 
shingles. 


1000 of shingles laid 4 inches to the weather are 
estimated to cover 109 square feet; laid 44inches to 
the weather, to cover 120 square feet ; and laid 5 
inches to the weather, to cover 133 square feet. 


For laying 1000 shingles carpenters allow about 6 pounds 
of 4-penny nails, or 5 pounds -’- venny nails. 

NUMBER OF SHINGLES IN A ROOF. First find the number of square 
inches in one side of the roof; cut off the right hand or unit figure, 
and the result will be the number of shingles required to cover both 
sides of the roof, laying five inches to the weather. The ridge-board 
provides for the double courses at the bottom. Illustration: Length 
of roof, 100 feet, width of one side, 30 feet, —190 x 30 x 44=—432,0'0. 
Cutting off the right-hand figure, we have 43,200 as the number of shin- 
gies required. 

CLAPBOARDS are usually 4 feet long, and are esti- 
mated by the 1000. 

100 of clapboards laid 4 inches to the weather are 
estimated to cover 1334 square feet ; laid 44 inches 
to the weather, to cover 150 square feet ; and laid 5 
inches to the weather, to cover 166% square feet. 


For laying 100 of clapboards carpenters allow about 34 


pounds of 5-penny nails. 


LaTHs are usually 4 feet long, 14 inches wide, and 
4 inch thick, and are put up in bundles containing 
100. 

1000 of laths, set 4 of an inch apart, are estimated 
to cover 55 square yards. 


For setting 1000 of laths carpenters allow about 7 pounds 
of 38-penny fine nails. 


116 


TABLE OF MULTIPLES, 


For the practical convenience of those who have occasion to refer 
to mensuration, we have arranged the following useful Table of Multi- 
ples. It covers the whole ground of practical Geometry. 


Diameter of a circle x 3.1416 = Circumference. 

Radius of a circle x 6.283185 = Circumference. 

Square of the radius of a circle x 3.1416 = Area. 

Square of the diameter of a circle x 0.7854 = Area. 

Square of the circumference of a circle x 0.07958 = Area. 

Half the circumference of a circle x by half its diameter = Area. 

Circumference of a circle x 0,159155 = Radius. 

Square root of the area of a circle x 0.56419 = Radius. 

Circumference of a circle x 0.31831 = Diameter. 

Square root of the area of a circle x 1.12839 = Diameter. 

Diameter of a circle x 0.86 = Side of inscribed equilateral triangle. 

Diameter of a circle x 0.7071 = Side of an inscribed square. 

Circumference of a circle x 0.226 = Side of an inscribed Square. 

Circumference of a circle x 0.282 = Side of an equal square. 

Diameter of a circle x 0.8862 = Side of an equal square. 

Base of a triangle x by ¥% the altitude — Area. 

Multiply both diameters and .7854 together = Area of an ellipse. 

Surface of a sphere x by 1-6 of its diameter = Solidity. 

Circumference of a sphere x by its diameter = Surface. 

Square of the diameter of a sphere x 3.1416 = Surface. 

Square of the circumference of a sphere x 0.3183 = Surface. 

Cube of the diameter of a sphere x 0.5236 = Solidity. 

Cube of the radius of a sphere x 4.1888 = Solidity. 

Cube of the circumference of a sphere x 0.016887 = Solidity. 

Square root of the surface of asphere x 0.56419 — Diameter. 

Square root of the surface of a sphere x 1.772454 — Circumference. 

Cube root of the solidity of a sphere x 1.2407 = Diameter. 

Cube root of the solidity of a sphere x 3.8978 — Circumference. 

Radius of a sphere x 1.1547 = Side of an inscribed cube. 

Sqare root of (1-3 of the square of) the diameter of a sphere = Side of 
inscribed cube. 

Area of its base x by 1-3 of its altitude= Solidity of a cone or pyramid 
whether round, square, or triangular. 

Area of one of its sides x 6= Surface of a cube. 

Altitude of trapezoid x 14 the sum of its parallel sides = Area. 


T7 


Paper Table for Frunters’ and Publishers’ Use, 


Showing the quantity of paper required for printing 1000 copies 
{including 56 extra copies to allow for wastage), of any usual sized 
Book from 8vo. down to 32mo. Lf the quantity required is not 
found in the Table, double or treble some suitable number of pages 


or quantity of paper. 


No. of | 8vo. | 12mo. | 16mo. | 24mo. | 32mo. 
Forms. | Pages. | Pages. | Pages. | Pages. | Pages. 


ef | | 


17 136 204 272 408 18 14 
18 144 216 288 432 19 16 
19 152 228 304 456 20 18 
20 160 240 | 320 | 480 22 

21 168 252 336 504 23 2 
22 176 264 | 352 2t 4 
23 184 276 368 Zone O 
24 192 288 384 20 8 
25 200 300 } 400 27,10 
26 208 312 | 416 28 12 
27 216 324 432 29 14 
28 224 336 448 50 16 
29 232 348 464 31 18 
30 240 360 480 33 

31 248 372 496 34 2 
32 256 384 512 BD 4 
38 264 396 528 3h «6 
34 272 408 544 37-8 
35 280 420 | 560 38 10 
36 288 432 576 39 12 
37 296 444 | 592 40 14 
38 304 456 608 41 16 
39 312 468 42 18 


118 


Names and Dimensions of Various 
Sizes of Paper. 


Pen iNede 


Medium. -+e+eee19 x 24 
Royal (20 x 24) .20 x 25 
Super Royal,....22 x 28 


Double Medium. 24 x 38 
Double Royal. ..26 x 40 
D’ble Super Roy’128 x 42 


Imperial, ....... 22 x 32 | D’ble Super Roy’l29 x 48 
Med’m and a half,24 x 30 | Broad Twelves..23 x 41 
Small Double Me- Double Imperial.32 x 46 

dium <--cves .24 x 36 

FOLDED. 
Billet Note .....- Gx. 1 Ltr «6 os Vie leslie 10 x 16 
Octavo Note..... 7x 9 | Commerc’ Letter 11 x 17 
Commercial Note 8 x 10 | Packet Post ....114x 18 
Packet Note..... 9x11 | Foolscap..... - -124x 16 
Bath Note.....-. 83x 14 
PIA T 

Legal Cap.--..-13 x 16 | Medium........18 x 23 
Flat Cap ...-... 14 x 17'| Royal....... 22019 x 24 
Crown .+.+----- 15x19 | Super Royal....20 x 28 
D’ble Flat Letter 16 x 20 | Imperial ...-...22 x 30 
Demy..+--+eee- 16 x 24 | Double Demy...21 x 81 
Folio Post...-.. 17 x 22 | Elephant ......224x273 
Check Folio ....17 x 24 | Columbier......23x31} 
Double Cap....-. 17x 28 | Atlas ...cee0e026 x 33 


Extra Size Folio.19 x 24 


Double Elephant 26 x 40 


119 


NAILS. 


The different sorts of nails are named either from the 
use to which they are applied, or from their shape, as 
shingle, floor, ship-carpenters’ and horse-shoe nails; rose- 
heads, diamonds, &c. The term penny, when used to 
mark the size of nails, is supposed to be a corruption of 
pound. Thus a four-penny nail was such that one thou- 
sand of them weighed four pounds, ten-penny such that 
one thousand weighed ten pounds, &c. 


RATES. 


Aithough nails are sold at various prices, according to 
the size. quotations called Mates give one price only, 
meaning that all sizes from 10d to 60d inclusive, sell for 
the rate named. For prices on other sizes add extra to 
the price named, according to the following table. 


EXTRAS. 

2d WinGi.cccccccscosssec ae $4.00 6d Floor see eeeeeseeeeseee ° $1.25 
Die ards crvediaras ale wis aiaieie's el tos O0 |) (no Gier GOH OOK ase<> ses Ropdaad) VANE 
2 INS dapeaiir Gop ooGor 6st 2.25] |10d Floor, and larger....... Py (3) 

Bereta ictces'c ose Se ins ei 1.50) | ———__$___—___—_——__ 
Ad Light. .a0) onsen ns tinas 1.50} | 6d Brad Head........ 1.50 
4a Swedes, com........ vetls (90500), (0 8a. AM hats Aad cua 1°25 
4a 66 Dar icetceces. 4) 4.00) 110d sé and larger...... 1.00 
4d & 5d) Common, ....-.-. 15 f 
6d & 74$ Fence and ‘ .50| | 8d Cooper & slate... ....: 2.00 
8d & 9d ) Sheathing. oe eo 4d. és AE eres ieee a my ds 
5d Hf J i yee Be 1.25 
TOS Ae hae Wa eae, os .25| | 6d “< Ce Series ie Meal ea TAO 
oleae - Seth aati Pha eine 15 
38d Fine Box.....-..+.- | hy 4.00) sro SPAT = 
SEs OX sacs siete aicleia.s wiels\als's.4/ S200} Lin HOON S F-ce).rs sere 3.00 
4d o6 ce cece eereeeeeseer- ee 2.0 a att a ee ee oe ee oe ee, —— 
BOih ae eaten. got me's Pt setatsto Leite eS teCLIN@is ce e)s.<icsierse = sa oe 1.00 
GAVGhHICL BOX. .as> == ecccece 1.00 SPSg APES ) MRSA Wal tea Camere 7 
BOTS OU Met Pete tien ain'en -(5| | Lin. Clinch ...-.......... 4.00 
10d Box, and larger. rere 5U| | yin. « Rear nt eres the 3 50 
os LEAS Vi ec Gis Sehr can ta ae 3.00 
4d Fine Finishing..... -.. 2.25} 14% I aie ewe ne vee 2.50 
5d ‘6 one 2.00 2& 2M, in} Clinchiesaee-ee: 2.00 
6d es 1 Perea 1.75] | 2144 & 2%, in. * Palsaleinewa | tle do 
8d Fe as tee 2.0788 1.50 3& 3% in. aa -o- eeee 1.50 
10d “6 TU Aweee |) 1.25) | Half Casks}‘additionaly 2% | \).25 
TRATES AIVO Ls ater ce he.c ceeds sae 2.50| | Galvanizing.. aisela poleiotets 2.50 
1i,zin. ‘6 2.2.00 wake eice's ZOO|p| SLINNING eign cts ctelwiere's o's SeSuIEGAO 


LENGTH OF NAILS. 
od 3d 4d 5d 6d 7d 8d 9d 10d 12d 20d 30d 40d 50d 60d 
lin. 14 1% 1% 2 2% 2% 2% 3 3% 4 4% 5 5% 6 
NUMBER OF NAILS IN A POUND. 


94 4d 5d 6d Td 8d 10d 12d 20d 30d 40d 50d 
557 353 232 167 141 101 68 54 34 16 12 10 


120 


HOW TO MEASURE LAND. 


Land can be measured with satisfactory accuracy 
for many purposes, by pacing. Five paces are equal 
to one lineal rod. A man having long legs usually 
measures more than a rod at five paces, while a short 
legged man will be obliged to step unnaturally long 
to measure a rod at five paces. The correct way is 
to measure 164 feet on level ground, then practice 
gauging the steps until one can measure one rod at 
every five steps, then one hundred steps or paces will 
be equal to twenty rods. Ifa plat of land be two 
hundred paces long and fifty paces wide, call every 
five paces a rod, multiply the rods in length by the 
rods in width, and divide the product by 160, the 
square rodsinanacre. Thus: 100 paces = 20 rods, 
and o0)*paces —=10erods3 10 920 —— 200G square 
rods, which, divided by 160, gives 14 acres. 

A square acre is about 208 feet 84 inches on ev- 
ery side. 

In order to lay out one acre of land four times as 


long as the width, the length must be 417 feet 5 
inches, and the width 104 feet 4 inches. 


Twenty feet front and 2,187 feet deep is one acre. 
Twenty-five feet front and 1,7424 feet deep is one 
acre. 


Thirty-three feet front and 1,320 feet deep is one 
acre. 


Forty feet front and 1,089 feet deep is one acre. 
Fifty feet front and 8764 feet deep is one acre. 


One hundred feet front and 4354 feet deep is one 
acre. 


In one square acre there are 48,560 superficial 
fect, 


121 
TRANSACTIONS WITH BANES. 


Make your deposits in the bank as early in the day 
as you conveniently can, and never without your bank- 
book. 


For your own security, it is well to have ONE PAR- 
TICULAR PERSON to do your business at the bank, 
who shall be competent to take charge of the money 
and papers you intrust to his care, and sufficiently in- 
telligent to understand and properly deliver the mes- 
sages and explanations you may have occasion to 
make ; also, that you write or stamp OVER YOUR IN- 
DORSEMENT, upon all checks which you send to be 
deposited to your credit in the bank, the words “ For 
DEPOSIT TO OUR CREDIT,” which will prevent their 
being used for any other purpose. 


Always use the deposit tickets furnished by the 
bank, and examine the date and indorsement of ev- 
ery check. When checks are deposited the banks 
require them to be indorsed by the depositor, whether 
drawn to his order or not. 


Keep your check-book, when not in use, under 
your own lock and key. Make it a rule to give 
checks only out of YOUR OWN CHECK-BOOK. 


Draw as few checks as possible. When you have 
several sums to pay, draw ONE CHECK for the whole, 
and take notes of such denominations as will enable 
you to distribute the amount among those you intend 
it for. 


Do not allow your bank-book to run too long with- 
out being balanced, and when returned by the bank 
compare it with your own account, and examine your 
cancelled checks without delay. If you wish to pre- 
serve your cancelled checks, deface or destroy the 
signature as soon as returned, in a manner that will 


122 


prevent their being copied, and place the checks out 
of the reach of others. 

In filling up checks, do not leave space in which 
the amount may be increased. It has been decided 
that when a check is so carelessly drawn that an al- 
teration may be easily made, the loss arising from 
the alteration, if any, must be borne by the drawer. 

Write your signature with your usual freedom, and 
never vary the style of it. 

Offer notes for discount or collection in good sea- 
son. Do not put off the offering of notes for dis- 
count until the last day of your need. When notes 
are discounted or collected for you, hand your bank- 
book to the clerk, that they may be entered in it to 
your credit. 


BROKERS’ TECHNICALITIES. 


A Butt is one who operates to depress the oe of 
stocks, that he may buy for a rise. : 

A Bear 1s one who sells stocks for future delivery, 
which he does not own at the time of sale. 

A CorRNER is when the Bears cannot buy or borrow the 
stock to deliver in fulfillment of their contracts. 

OVERLOADED is when the Bulls cannot take and pay for 
the stock they have purchased. 

SHorT is when a person or party sells stocks when they 
have none, and expect to buy or borrow in time to deliver. 

LonG is when a person or party has a plentiful supply 
of stocks. 

A Poot or RING is a coinbination formed to control the 
price of stocks. 

A broker is said to Carry stock for a customer when he 
has bought and is holding it for his account. 

A Wash is a pretended sale by special agreement be- 
tween buyer and seller, for the purpose of getting a quota- 
‘ion reported. 

A Pur AND CALL is when a person gives so much per 
cent. for the option of buying or selling so much stock on 
a certain fixed day, at a price fixed the day the option is 
given. 


MARKING GOODS. 


In buymg goods the merchant is often at a loss: 
to know whether the price of the article suits his 
market or not; ard if he is not a good accountant 
it often takes him some time to determine. Those 
who buy largely can best appreciate the value of a 
short method of calculating the percentage desired. 

If you wish to calculate the per cent. on a single 
article, the following is considered the best method. 
If you wish to sell an article at any of the following 
per cents., say the article cost 70 cents, and you 


wish to make 


10% Divide by 10, Multiply by 1177. 
20 « “10, “19-84, 
25 ‘* Multiply by 10, Divide “ 887%. 
30 * Divide by.10, Multiply “© 18=91, 
834 Add 4 of itself. = 934. 
334% Divide by 3, Multiply by 4=-933. 
50 “ Add 4 of itself. ==1.05. 


124 


Another method of marking 25 per cent. profit is 
to cut off the right-hand figure, and you have the 
price in shillings and pence: thus, if you buy an 
article for 60 cents, and wish to gain 25 per cent., 
cut off the right-hand figure and you have 6 shillings 
of 124 cents each, or 75 cents, the cost with 25 per 
cent added. 

If the figure you cut off is not a cipher, add 4, 
thus: 3 cents, add #; 5 cents, 14; 6 cents, 14, etc. 

EXAMPLE.—Suppose an article cost 74 cents, and 
you wish to make it 25 per cent. advance, cut off the 
right-hand figure and you have 7s.4d., 4=$=—1, 
added to 4 = 5, 7s. = 874 +5 = 925 cents. 


How to mark an article bought by the dozen, to 
taake 20 per cent. 


Remove the decimal point one place to the left. 


EXAMPLE.—Suppose a lot of hats cost $2.50 per 
dozen, by removing the decimal point one place to 
the left we have 20 per cent. and cost, or .25 
apiece for the hats. To ascertain any other per 
cent., we take the basis at 20 per cent. and add or 
subtract, as the case may be. 

To make 25 per cent., remove the point one place | 
to the left, and add ,}. 


To make 30 % add 7, itself. 


66 334 6 6 1 66 
66 85 6 8s 4 66 
66 384 66 6G 4 rT 
66 40 “ «& 3 66 
66 44 6 6 1 66 
66 50 “é 66 4 66 
fs OO Se es 

‘ 80 « « 3 re 


These additions must be made after removing the 
point as above directed, and this sum will always be 
the selling price of a single article. 


125 
The above table contains all the per cents. gener- 
ally used in business, and can easily be applied. 


This rule is very valuable to the merchant in buy- 
ing goods; suppose he buys his goods at auction, he 
does not have sufficient time to make extensive cal- 
culations before the goods are cried off. But by 
knowing that at 20 per cent. profit, he need not 
change a figure, he can tell instantly whether he can 
afford to buy those goods or not. 


MARKING GOODS. 


It is customary for merchants to have a private 
mark, denoting the cost and often the selling price. 
These marks are sometimes made up of peculiar 
characters, but mostly letters of the alphabet that 
represent the nine digits. For example: 


BLACK HORSE 
1234567890 


Suppose an article cost $2.25, and you wish to 
sell it for $5.00, the mark would be thus: yi 


Usually they have what is called a repeater, that 
is to be used where a letter is repeated, as above. 
suppose G to be the repeater; then instead of 
using the letters L and E twice, we insert the repeat- 
er, thus: j&%. It sometimes happens that there are 
but two letters in the cost price and three in the 
selling price; to avoid this, place the letter repre 


senting 0 as the first letter in the cost, thus: write 


eok 


75 cents cost. 1.00 selling price : fee. 


HOW TO TELL» 


THE DAY OF THE WEEK, 
THE DAY OF THE MONTH, 
THE MONTH IN THE YEAR, 


THE AGE IN YEARS, 
WITHOUT ASKING A SINGLE QUESTION, 


PROCESS. 


Ask the person you wish to figure out the above 
facts for, to write down first, the day of the week on 
which he or she was born; if this is not known, 
ascertain by preceding method ; next, the day of the 
month, next, the month in the year, then multiply 
the whole by 2, add 5, multiply by 50, add age, sub- 
tract 3865, add 115, the result will be ; the first figure 
will be the day of the week ; the next, the day of the. 
month ; the next the month in the year, and the last 


the age in years. 


127 


ExXaMPLE.—I was born Wednesday, May 31, 1843. 

Process.—Ist. Write 4 as the first figure, because 
Wednesday is the 4th day of the week. 

2d. Write 5 as the second figure, because May is the 5th 
month. 

3d. Write 31 as the 3d and 4th figures, because this is 
the day on which I was born. ‘The figures therefore read, 

ety rast 
2 multiply. 


5 0 multiply. 


453350 

4 1 add age. 
4533891 

3 6 5 subtract. 
453026 

1 1 5 add 
4,6,3 1.4 1 
Beis 
27 Sf 
as & 
< B 
a 


ExAMPLE 2d.—A friend was born April 6th. 
Take number of month and day of month. 
5 6 


2 multiply. 
5 added. 
hy J 
5 0 multiply. 
add age. 
90 0 
3 6 5 subtract. 


5 
5 add. 


HOW 10 TELL THE DAT OF THE WEEE, 


A scientific method of telling immediately what 
day of the week any date transpired or will tran- 
spire, from the commencement of the Christian Era 


for the term of three thousand years. 


MONTHLY TABLE. 


The ratio to add for each month will be found in 
the following table : 


Ratio of June is....-..... 0 | Ratio of October is......- 3 
Ratio of September is-.-.1 | Ratio of May is..-..-. oafald, 
Ratio of December is-..-. 1 , Ratio of August is.-...... 5 
Ratio of April is......... 2 | Ratio of March is........ 6 
Ratio of July is ..... ....2 | Ratio of February is......6 
Ratio of January is....... 3 | Ratio of November is.---6 


Notre.—On Leap Year the Ratio of January is 2, and 
the ratio of February is 5. The ratio of the other ten 


months do not change on Leap Years. 


CENTENNIAL TABLE. 


The ratio to add for each century will be found in~ 
the following table : 


, 200, 900, 1800, 2200, 2600, 3000, ratio is. ....-.-- 0 
E 800, 1000, oes 5 iesss | seus ieee  TALIO IS tensemee sO 
& 400, 1100, 1900, 2800, 2700, ---. ratio is. ........ 5 
= 500, 1200, 1600, 2000, 2400, 2800 ratio is. ........ 4 
EF (600; 1300), ss" Weeds hk Wee agtetien Hobie yee tienes 3 

000, 700. 1400, 1700 2100 2500 2900 ratio is. ....-.-- 2 


1002800) )2500,) sins) Ae ie ALTO aletam eee area d 


Notr.—The figure opposite each century is its ratio; 
thus the ratio for 200, 900, etc., is 0. To find the ratio of 
any century, first find the century in the above table then 
run the eye along the line until you arrive at the end; the 


small figure at the end is its ratio. 


129 


METHOD OF OPEKATION. 


Ru e.*—To the given year add its fourth part, rejecting 
the fractions; to this sum add the day of the month; then 
add the ratio of the month and the ratio of the century. 
Divide this sum by 7; the remainder is the day of the 
week, counting Sunday as the first, Monday as the second, 
Tuesday as the third, Wednesday as the fourth, Thursday 
as the fifth, Friday as the sixth, Saturday as the seventh; 
the remainder for Saturday will be 0 or zero. 


EXAMPLE 1.—Required the day of the week for 
the 4th of July, 1810. 


To the given year, which is.........s.. eo sevesecess 10 

Add its fourth part, rejecting fractions....... ern ete oie 2 

Now add the day of the month, which is.........-- - 4 

Now add the ratio of July. which is....-....++e2e--- 2 

Now add the ratio of 1800, which is..... Bt cietsete ol) eis 0 

Divide the whole sum by 7. 7 | 18—4 
2 


We have 4 for a remainder which signifies the 
fourth day of the week, or Wednesday. 


Norz.—In finding the day of the week for the present 
century, no attention need be paid to the centennial ratio, 
as it is 0. 


EXAMPLE 2.—Required the day of the week for 
the 2d of June, 1805. 


To the given year, which is.......... iat acaak ese 5 

Add its fourth part, rejecting fractions..........- +. 1 

Now add the day of the month. which is........... ~ 2 

Now add the ratio of June, which is.++--++s.ee++ eee. 0 

Divide the whole sum by 7. 7| 8—1 
1 


We have 1 for a remainder, which signifies the 
first day of the week, or Sunday. 

The Declaration of American Independence was 
signed July 4, 1776. Required the day of the week. 


“* When dividing the year by 4, always leave off the centuries. We 
a.vide by 4 to find the number of Leap Years. 


i 


130 


To the given year, which is.......esecesecceceereee IO 
Add its fourth part, rejecting fractions...- sese++---19 
Now add the day of the month, which is ....e+-e+++ 4 


Now add the ratio of July, which is...... S wenn ta daw 

Now add the ratio of 1700, which is.....--.eee+eeees 2 

Divide the whole sum by 7, 7 | 1083—5 
14 


We have 5 for a remainder, which signifies the 
fifth day of the week, or Thursday. 

The Pilgrim Fathers landed on Plymouth Rock 
Dec. 20, 1620. Required the day of the week. 


To the given year, which is.......- a ejnlete's Raima eee 20 

Add its fourth part, rejecting fractions .---..-.....-- 5 

Now add the day of the month, which is.........--.20 

Now add the ratio of December, which is ..--.-.ee«- 1 

Now add the ratio of 1600, which is.......-cseee ey: | 

Divide the whole by 7, 7 | 50—1 
7 


We have 1 for a remainder, which signifies the 
first day of the week, or Sundav. 


AMUSING ARITHMETIC. 


Under the head of Amusing Arithmetic we give a 
collection of problems particularly adapted to the 
social circle, or the fireside, of a winter evening. 
The most of those problems are in the form of puz- 
zles, and some of them particularly amusing, The 
majority of them are very old, their parentage being 
entirely unknown, so that no credit can be given to 
their authors. This is believed to be the largest 
collection ever published. 


1. Think of a number of 3 or more figures, divide 
by 9, and name the remainder; erase one figure of 
the number, divide by 9, and tell me the remainder, 
and I will tell you what figure you erased. 

\TETHOD. --If the second remainder is less than -the first, 
the figure erased is the difference between the remainders; 
but if the second remainder is greater than the first, the 
figure erased equals 9, minus the difference of the re- 
mainders. 

2, Think of a number, multiply it by 8, and mul- 
tiply it also by 4, take the sum of the squares of the 
products, extract the square root of this sum, divide 
by the first number, and I will name the quotient. 

MertTHop.—The quotient will always be 5. The same 
will be also true if we have them multiply and divide by 
the same multiples of 3, 4, and 5, as 6, 8, 10, &c. If we 
have them divide by 5, it will give the number they com- 
menced with. 

3. Think of a number, multiply it by 5, also by 
12 ; square each product, take their sum, extract the 
square root, divide by the number commenced with, 
and I will name the quotient. 

MetuHop.—The quotient is always 13. To give variety it 


is well to use multiples of 5,12; as 10, 24, &c., and then 
the quotient is 26, &c. 


132 


AMUSING ARITHMETIC. 


4, Think of a number composed of two unequal 
digits, invert the digits, take the difference between 
this and the original number, name one of the digits 
and I will name the other. 

MeTHop.—The sum of the digits in the difference is al- 
ways 9; hence when one is named, the other equals 9 
minus the one named. 

5. Take any number consisting of three consecutive 
digits and permutate them, making 6 numbers, and 
take the sum of these numbers, divide by 6, and tell 
me the result, and I will tell you the digits of the 
number taken. 

METHOD.—The quotient consists of three equal digits; 
the digits of the number taken are, Ist, one of these equal 
digits; 2d, this digit increased by a unit; 3d, this digit 
diminished by a unit. The same principle holds when 
the digits of the number taken differ by 2, 3, or 4. It is a 
very pretty problem to prove that the sum is always divis- 
ible by 9, and 18. 

6. Think of a number greater than 3, multiply it 
by 3; if even, divide it by 2; if odd, add 1, and then 
divide by two. Multiply the quotient by 3; if 
even, divide by 2; if odd, add 1, and then divide by 
2. Now divide by 9 and tell the quotient, without 
the remainder, and I will tell you the number 
thought of, 

MeETHuHoD. — If evex both times, multiply the quotient by 
4; if even 2d, and odd lst, multiply by 4, and add 1; if 
even Ist, and odd 2d, multiply by 4, and add 2; if odd 
beth times, multiply by 4, and add 3. 


7. Take any number, divide it by 9 and name the 
remainder. Multiply the number by some number 
which I name, and divide this product by 9, and I 
will name the remainder. 


MeETHOD.—To tell the remainder, I multiply the first 
remainder by the number by which I told them to multi- ° 
ply the given number, and divide this product by 9. ‘The 
remainder is the second number that they obtained. 


133 


AMUSING ARITHMETIC. 


8. A and Bhave an 8 gallon cask full of wine, 
which they wish to divide into two equal parts, and 
the only measures they have are a 5 gallon cask and 
a 3 gallon cask. How shall they make the division 
with these two vessels? 

MetTuop.—Fill the 3 and pour it into the 5, then fill it 
again, and from it fill up the 5, which will leave one gal- 
lon in the 3 gallon keg; empty the 5 in to the 8, and pour 
the one from the 3 into the 5; fill the 8 again and empty 
into the 5; then there are four gallons in the 5 gallon keg, 
and the same left in the 8. 


9. Two men have 24 ounces of fluid, which they 
wish to divide between them equally. How shall 
they effect the division, provided they have only 
three vessels , one containing 95 ozs., the other 11 
ozs., and the third 13 ozs. 


10. Two men, stopping at an oyster saloon, laid a 
wager as to which could eat the most oysters. One 
eat ninety-nine, and the other eat a hundred and 
won. How many did both eat? 


REMARK.—The ‘‘catch” is in ‘‘a hundred and won.” 
When this is repeated it sounds as if it meant ‘‘one eat 
99 and the other eat 101;” hence the result usually given is 
200. Thecorrect result, of course, is 199. 


11. Six ears of corn are in a hollow stump. How 
long will it take a squirrel to carry them all out, if he 
takes out three ears a day? 


ReMARK.—The ‘‘catch” is in the word ears. He car- 
ries out two ears on his head, and one ear of corn each 
day; hence it will take him 6 days. 


12. A and B went to market with 30 pigs each, 
A sold his at 2 for $1, and B at the rate of 3 for $1, 
and they, together, received $25. The next day A 
went to market alone with sixty pigs, and, wishing 


134 


AMUSING ARITHMETIC. 


to sell at the same rate, sold them at 5 for $2, and 
received only $24. Why should he not receive as 
much as when B owned half of the pigs ? 
MetHop—The insinuation that the first lot were sold at 
the rate of 5 for $2. being only true in part. They com- 
mence selling at that rate, but after making 10 sales, A’s 
pigs are exhausted, and they have received $20.; B still 
has 10, which he sells at ‘‘2 for a dollar,” and of course 


receives $5; whereas, had he sold them at the rate of 5 for 
$2, he would have received but $4. 


13, In the bottom of a well, 45 feet deep, there 
was a frog which commenced travelling towards the 
top. In his journey he ascended 3 feet every day, 
but fell back two feet every night. In how many 
days did he get out of the well? 


14. A man having a fox, a goose, and some corn, 
came to a river which it was necessary to cross. He 
could, however, take only ove across at a time, and 
if he left the goose and corn, while he took the fox 
over, the goose would eat the corn; but if he left 
the fox and goose, the fox would kill the Boos 
How shall he get them all safely over? 


Mrtuop.—Let him first take over the goose, leaving the 
fox and the corn, then let him take over the fox, and bring 
back the goose, then take over the corn, and lastly, take 
over the goose again. 


15. A man went to a store and purchased a pair 
of boots worth $5, and hands out a $50 bill to pay 
for them ; the merchant, not being able to make the 
change, passes over the street to a broker and gets 
the bill changed, and then returns and gives the 
man who bought the boots his change. After the 
purchaser of the boots has been gone a few hours, 
the broker, finding the bill to be a counterfeit, re- 
turns and demands $50 of good money from the 


135 


AMUSING ARITHMETIC. 


merchant. How much did the merchant lose by the 
operation. 

REMARK.— At first glance some say $45 and the boots; 
some, $50 and the boots; some, $95 and the boots; and 
others, $100 and the boots. Which is correct? 

16. What relation to me is my mother’s brother- 
in-law’s brother, provided he has but one brother ? 


17. Three men, travelling with their wives, came 
to a river which they wished to cross. There was 
but one boat, and but two could cross at one time; 
and, since the husbands were jealous, no woman 
could be with a man unless her own husband was 
present. In what manner did they get across the 
river. 

Metuop.—Let A and wife go over, let A return, let B’s 
and C's wives go over, A’s wife returns, B and C go over, 
B and wife return, A and B go over, C’s wife returns, and 
A’s and B’s wives go over, then C comes back for his wife. 

Simple as this question may appear, it is found in the 
works of Alcuin who flourished a thousand years ago; 
hundreds of years before the art of printing was invented. 
—Parke. 

18. Suppose it were possible for a man, in Cincin- 
nati, to start on Sunday noon, when the sun is in the 
meridian, and travel westward with the sun, so that 
it might be in his meridian all the time. He would 
arrive at Cincinnati next day at noon. Now, it was 
Sunday noon when he started, it has been noon with 
him all the way around, and is Monday noon when 
he returns. The question is at what point did it 
change from Sunday noon to Monday noon? 


19. Suppose a hare is 10 rods before a hound, and 
that the hound runs 10 rods while the hare runs 1 
rod. Now when the hound has run the 10 rods, the 
hare has run 1 rod; hence they are now one rod 
apart, and when the hound has run that 1 rod, the 


136 


AMUSING ARITHMETIC. 


hare has run 4, of a rod; hence they are now 7 of 
a rod apart, and when the hound has run the 7; of 
a rod they are z§, of a rod apart; and in the same 
way it may be shown the hare is always +5 of the 
previous distance ahead of the hound; hence the 
hound can never catch the hare. How is the con- 
trary shown mathematically. 


20. Think of any three numbers less than 10. 
Multiply the first by 2, and add 5 to the product. 
Multiply this sum by 5, and add the second number 
to the product. Multiply this last result by 10, and 
add the third number to the product; then subtract 
250. Name the remainder, and J will name the 
numbers thought of, and in the order in which they 
were thought of. 

Mertruop.—The three digits composing this remainder 
will be the numbers thought of; and the order in which 


they were thought of will be the order of hundreds, tens, 
and units. 


21. Write 24 with three equal figures, neither of 
them being 8. 


METHOD.—22 + 2 = 24, or 3? —3 = 24: 


22. Put down four marks, and then require a 
person to put down five more marks, and make ten. 
Metuop.—The four marks are as represented 
in the margin; the five more, making ten, are | | | | 
placed as in the margin. TEN 
23. Which is the greater, and how much, six 
dozen dozen, or one-half a dozen dozen, or is there 
no difference between them? 


24. Show what is wrong in the following reason- 
ing:—8—8 equals 2— 2; dividing both these 
equals by 2— 2 and the result must be equal ; 8—-8 
divided by 2 —2 = 4, and 2—2 divided by 2—2 


137 


AMUSING ARITHMETIC. 


= 1; therefore, since the quotients of equals divid- 
ed by equals, must be equal, 4 must be equal to 1. 

25. A man has a triangular lot of land, the largest 
side being 156 rods, and each of the other sides 68 
rods ; required the value of the grass on it, at the 
rate of $10 an acre. 

ReMARrK.—The ‘‘ catch ’ in this is, that the sides given 
will form no triangle. 

26. Says A to B, ‘Give me four weights, and I 
can weigh any number of pounds not exceeding 40.” 
Required the weights and method of weighing. 

ANSWER.—The weights are J, 3,9, and 27 pounds. In 
weighing, we must put one or more in both scales, or 
some in one scale and some in the other; thus, 7 lbs. = 
9 lbs. +1 lb. —3 Ibs. 

27. Mr. Frantz planted 13 trees in his garden, in 
such a manner that there were 12 rows. and only 3 
trees‘ in each row. In what manner were they 
planted ? 

ANSWER.—They were in the form of a regular hexagon, 
having a tree in the centre, and one at the middle and ex- 
tremity of each side. 


28. A and B raised 749 bushels of potatoes on 
shares ; A was to have #, and B # of them. Before 
they were divided, however, since A had used 
49 bushels, B took 28 bushels from the heap, and 
then divided the remainder according to the above 
agreement. Was this division fair? if not, show 
how it should have been. 


29. Two-thirds of six is nine, one-half of twelve is 
Ae SEVEN, 
The half of five is four, and six is half of eleven. 


So.ution.--Two thirds of §]X is |X; the upper half 


of X|| is Vii; the half of FEWE is [Y; and the upper 
half of X] is VI. 


138 


AMUSING ARITHMETIC. 


30. Does the top of a carriage wheel move faster 
than the bottom? 

METHOD. —This seems absurd, but it is strictly true, as 
any one may Satisfy himself in a moment, by setting up a 
stake by the side of a wheel, and move the wheel forward 
a few inches. 


31. Supposing there are more persons in the 
world than anyone has hairs on his head, there 
must be, at least, two persons who have the same 
number of hairs on the head, to a hair. Show how 
this is. 

32. Place 17 little sticks — 
matches, for instance — making 6 | 
equal squares, as in the margin. 

Then remove 5 sticks, and leave 3 | | 
perfect squares of the same size. == 


33. Three persons own 51 quarts of rice, and 
have only two measures; oné a 4 quart and the 
other a7 quart measure. How shall they divide it 
into three equal parts? 

MeEtTHOD.—Perhaps the easiest way is to give each one- 
17 quarts, which may be obtained thus: fill the 7 quart 
measure, empty this into the 4 quart measure, and there 
will be 3 quarts in the 7 quart measure, which added ta 
two 7 quart measures, equals 17 quarts. 


34, What four United States coins will amount to 
fifty-one cents? 


ANSWER.—Two 25 ct. pieces and two half-cents. 


35. How may the nine digits be 
arranged in a rectangular form, so | 4 | 9 | 2 
that the sum of any row, whether | | 
horizontal, vertical, or diagonal, 
5? 
shall equal 10: A en oe 


AnsweErR.—As in the margin. 


139 


AMUSING ARITHMETIC. 


36. How may the first 16 | 1 | 16/111] 6 
digits be arranged, so that the 
sum of the vertical, and hor- | i138 | 4 | 7 | 10 
izontal, and the two oblique 
rows, may equal 34? 

ANSWER.—As in the margin. 19°05 oN ay 


37. In what manner 
may the-first 20 digits, bey) eis eb ie 
arranged, sothatthe sum | g | 31 | 90 | 22 | 3 
of each row of five fig- |———|——_|——-|_|__ 
ures may be 65? 1s Rl i -  R 


om | |) 


ANSWER.—“*As in the mar- 
gin. PREP OUT RS ORS Ce adh (eit 
REMARK.—The above are | 95 | 9 Sn TAL Org 
called Magic Squares. They 
are very interesting, and have engaged the attention of 
some of our greatest mathematicians, among whom we 
may mention Leibnitz, Stifels, &c. The methods of ar- 
rangement given above are by no means the only ones 
that may be used. For the second problem, Frenicle, a 
French mathematician, has shown that there may be 878 
different arrangements. 


38. Take 10 pieces of money, lay them in a row, 
and require some one to put them together in heaps 
2 in each, by passing each piece over 2 others. 

MetTuHop.—Let the pieces be represented by the numbers 
bets ob4n 0507 44°) 95) LUs. bace 7,on 10,) Don 2. Sy Ons; 
1 on 4, and 9 on 6. 

39. An old Jew took a diamond cross to a jeweller, 
to have the diamonds reset; and fearing that the 
jeweller might be dishonest, he counted the dia- 
monds, and found that they numbered 7, in three 
different ways. Now the jeweller stole two dia- 
monds, but arranged the remainder so that they 
counted 7 each way. as before. How was it done? 


140 


AMUSING ARITHMETIC, 


MrtTHop.—The form of the cross when  yy.y. wg. 
left is represented by Fig. 1, and when re- 7 7 


turned by Fig. 2. It will be seen by the fig- wenee fet 

ures how the diamonds were counted by the 4 4 
3 os 

old Jew, and how they were arranged by the 2 2 
i i 


jeweller, who ‘‘jewed ” the Jew. 


40. Let a person select a number greater than 1 
and not exceeding 10; I will add to it a number not 
exceeding 10, alternately with himself ; and, although 
he has the advantage in selecting the number to start 
with, I will reach the even hundred first. 

METHOD.—I make my additions so that the sums are, 
respectively, 12, 23, 34, 45, &c., to 89, when it is evident 
Ican reach the hundred first. With one who does not 
mistrust the method, I need not run through the entire 
series, but merely aim for 89, or, when the secret of this is 
seen, for 78, then 67, &c. 

41, Let a person think of any number on the dial- 
face of a watch ; I will then point to various num- 
bers, and at each he will silently add ove to the num- 
ber selected until he arrives at ¢wenty, which he will 
announce aloud, and my pointer will be upon the 
number he selected. 

MetHop.—I point promiscuously about the face of the 
watch until the eighth point, which should be upon “ 12;” 
and then pass regularly around, towards ‘‘1,” pointing at 
611,” $10,” °9,” &c., until ‘‘twenty is called, when, as 
may be easily shown, my pointer will be over the number 
selected. 

42. Is there any difference between the results of 
the two following problems, and if so, what is it? If 
the half of 6 be 3, what will the fourth of 20 be? If 
3 be the half of 6, what will be the fourth of 20? 


43. A vessel with a crew of 30 men, half of whom 
were black, became short of provisions ; and, fearing 
that unless half the crew were thrown overboard, all 
would perish, the captain proposed to the sailors ta 


141 


AMUSING ARITHMETIC, 


stand upon deck in a row, and every ninth man be 
thrown overboard, until half the crew were destroyed. 
It so happened that the whites were saved. Re- 
quired the order of arrangement. 

ANSWER.—W. W. W. W. B. B. B. B. B. W. W. B. W. 
W. W. B. W. B. B. W. W. B. B. B. W. B. B. W. W. B. 

This can easily be found by trial, using letters or figures 
to represent the men. 

44, Think of a number, multiply it by 6, divide 
this product by 2, multiply by 4, divide by 3, add 40, 
divide by 4, subtract the number thought of, divide 
by 2, and the quotient is 5. Show why this is so. 


45, If through passenger trains, running to and 
from Philadelphia and San Francisco daily, start at 
the same hour from each place (difference of longi- 
tude not being considered ) and take the same time, 
six days, for the trip, how many through trains will 
the Pacific Express that leaves the San Francisco 
depot at 9 p. M., Sunday, have met when it reaches 
the Philadelphia depot? 


46. A switch siding to a single track railroad is 
just long enough to clear a train of eight cars and a 
locomotive. How can two trains of sixteen cars 
and a locomotive each, going in opposite directions, 
pass each other at this siding, and each locomotive 
remain with and have the same relative position to 
its own train after as before passing? 


47. Two hunters killed a deer, and sold it by the 
pound in the woods. ‘They had no proper means of 
weighing it, but knowing their own weights—one 130 
pounds and the other 190 pounds they placed a 
rail across a fence so that it balanced with one on 
each end. They then exchanged places, and the 
lighter man taking the deer in his lap, the rail again 
balanced. What was the weight of the deer ? 


142 


AMUSING ARITHMETIC. 


VARIATIONS. 


The doctrine of variations and combinations forms 
the basis of many forms of Lotteries, and of other 
calculations used in practical life. We shall com- 
mence with the simplest form of variations in which 
all the articles are taken at once and which is called 


PERMUTATION. 


To determine the number of permutations, com- 
mence with unity and multiply by the successive 
terms of the natural series 1, 2, 8, &c., until the 
highest multiplier shall express the number of indi- 
vidual things. The last product will indicate the 
number of possible changes. 


Example 1. How many changes can be made in 
the arrangement of 5 grains of corn, all of differ- 
ent colors, laid in a row? 

SOLUTION. —1X2X38X4X5=120, Axs. 


This may seem improbable, the number being so great, 
but if there were but a single grain more, the possible 
changes would be 720; and another would extend the 
limit to 5010; and so onward in a constantly increasing 
ratio. The reason, however, will be obvious on a little 
scrutiny. If there weré but one thing, as a, it would ad- 
mit of but one position; but if two, as a 6, it would admit 
of two positions, ad, da. If three things, as adc, then 
they will admit of 1X2X3=6 changes, for tne last two will 
admit of two variations, as adc, acé, and each of the 
three may successively be placed first, and two changes 
made to each of the others, so that 83X2=6, the number of 
possible changes. In the same way we may show that if 
there be four individual things, each one will be first in 
each of the six changes which the other three will under- 
go, and consequently, there will be 24 changes in all. In 
this way we might show that when there are 5 individual 
things, there will be 5 times as many changes as when 
there were but 4; and when 6, there will be 6 times as 


1438 


AMUSING ARITHMETIC, 


many changes as when there are only 5; and so on ad 
infinitum, according to the same law. 

Example, 2. In how many ways may a family of 
10 persons seat themselves differently at dinner? 

ANSWER.—3628800. 

When we consider that this would require a period of 
99353, years, the mind is lost in astonishment. The 
story of the man who bought a horse at a farthing for the 
first nail in his shoe, a penny for the second, &c., is thrown 
into the shade; and we incline to doubt whether there is 
not some mistake; and yet on just such chances as one to 
all these do gamblers constantly risk their money! 

Example 3. 1 have written the letters contained 
in the word NIMROD on 6 cards, being one letter 
on each, and having thrown them confusedly ‘into a 
hat, I am offered $10 to draw the cards successively, 
so as to spell the name correctly. What is my 
chance of success worth? 

ANSWER. —1,% cents. 

48. Sold a horse for $56, and gained as much per 
cent. as the horse cost me. Required the cost. 

Roize.—‘‘ Multiply the selling price by 190, and add 2500 
to the product; of the sum extract the square root, and 
from the root subtract 50. The remainder will be the 
prime cost.” 


Horse sold for $56 Proof— 
100 Cost $40 
a Percent. 40 
5600 — 
+2500 Gain 16.00 
Cost 40 
¥8100—90 — 


—50 Sold for $56 


Leaves $40 cost. 

Human ingenuity would perhaps fail to find a reason 
for the above rule, by the aid of common arithmetic mere- 
ly, or to explain the steps satisfactorily to a learner. It 
seems to be without reason, and yet it will solve all ques- 
tions involving a similar principle. 


144 


AMUSING ARITHMETIC. 


49. Suppose 2000 soldiers had been supplied with 
bread sufficient to last them 12 weeks, allowing each 
man 14 ozs. per day; but on examination they find 
105 barrels, containing 200 lbs. each, wholly spoiled ; 
what must be the allowance to each man, that the 

remainder may last them the contemplated time? 

ANALYSIS, as follows :—I1st. If one man ate 14 ozs. in a 
day, he would eat 7 times 1498 ozs. in a week; and if he 
ate 98 ozs. in a week, he would eat 12X98=1176 ozs. in 12 
weeks; and 2000 men would eat 2000X1176 ozs.=2352000 
ozs. in 12 weeks. 

105 barrels of 200 lbs. each=21000 lbs. destroyed; and 
21000X16=836000 ozs., which deducted from the whole 
quantity 2352000 ozs. leaves 2016000 ozs. to be consumed. 
Then if 2000 men consume 2016000 ozs. in 12 weeks, 1 man 
will consume 2918°9°%==1008, ozs.; and if 1 man consume 
1008 ozs. in 12 weeks, he will consume 199884 ozs. in 1 
week, and 8412 ozs. inl day. Azs. 


50. How may 100 be expressed with four nines? 

ANSWER. — 9932. 

51. What three figures, multiplied by 4, will make 

precisely 5? | 

ANSWER. — 11, or 1.25. 

52. Required to subtract 45 from 45 and leave 

45 as a remainder. 

SOLUTION. — 9-+8--7-++-6-+-5+4+3121 145 
11243441516174819—45 
8-+6+4-+-1+9-+7-+5+3-2—=45 

53. From 6 take 9; from 9 take 10; 

From 40 take 50, and 6 will remain! 


SoLu110N.— SIX IX XL 
IX ,¢ 1 


S I X 


The majority of these puzzles and problems being 
founded upon principles quite easily comprehended, 
itis not thought necessary to explain the principles 
of the puzzles nor solve the problems. it is hoped 
that they may prove a source of pleasure and profit. 


fem 


3 0112 017102598 


